{
"cells": [
{
"cell_type": "markdown",
"id": "a77c1997-ce07-4d7f-96a3-a8a7811aa511",
"metadata": {},
"source": [
"# Difference Finite Scheme derived from MRT Lattice Boltzmann for Convective-Diffusive Equation"
]
},
{
"cell_type": "markdown",
"id": "b23f99a1-bc9b-4657-9fe8-a46c34b89270",
"metadata": {},
"source": [
"Based on the works of {cite:t}`bellotti2022finite` and {cite:t}`chen2023fourth`, we will develop a difference finite description of lattice Boltzmann equation in the multi-relaxation-times (MRT) framework."
]
},
{
"cell_type": "markdown",
"id": "88580bae-c33c-41ef-b735-ff704777543f",
"metadata": {},
"source": [
"## MRT Lattice Boltzmann equation for Convective-Diffusive Equation"
]
},
{
"cell_type": "markdown",
"id": "963d0b5e-c67e-4c54-b581-b49874e600c7",
"metadata": {},
"source": [
"\n",
"MRT Lattice Boltzmann equation for Convective-Diffusive Equation\n",
"
"
]
},
{
"cell_type": "markdown",
"id": "ce1deef2-8ab2-48f3-bde8-cdd328bac60e",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": []
},
"source": [
"To describe convective-diffusive problems, the MRT lattice Boltzmann formulation is described by:\n",
"\n",
"$$\n",
"f_i( x_{\\alpha} + e_{i,\\alpha} \\delta t, t+\\delta t) = f_i(x_{\\alpha}, t) - (\\textbf{M}^{-1}\\textbf{S}\\textbf{M})_{ik}\\left( f_{k} - f_{k}^{eq} \\right), \n",
"$$(MRT-LB-conv-df-Eq)\n",
"\n",
"where $\\textbf{M}$ is matrix of moments, $\\textbf{S}$ is the diagonal matrix of relaxation times, and $\\textbf{M}^{-1}$ is the inverse matrix of $\\textbf{M}$. Considering the problem description in a lattice D1Q3, we have:\n",
"\n",
"$$\n",
"\\textbf{M} =\\begin{pmatrix} 1 \\\\ e_{i,x} \\\\ 3e_{i,x}^{2}-2 \\end{pmatrix} =\\begin{pmatrix} 1 & 1 & 1 \\\\ 0 & 1 & -1 \\\\ -2 & 1 & 1 \\end{pmatrix}, \\qquad \\qquad \\textbf{S} = \\begin{pmatrix} s_0 & 0 & 0 \\\\ 0 & s_1 & 0 \\\\ 0 & 0 & s_2 \\end{pmatrix}, \\qquad \\qquad \\textbf{M}^{-1}=\\begin{pmatrix}\\dfrac{1}{3} & 0 & -\\dfrac{1}{3} \\\\ \\dfrac{1}{3} & \\dfrac{1}{2} & \\dfrac{1}{6} \\\\ \\dfrac{1}{3} & -\\dfrac{1}{2} & \\dfrac{1}{6} \\end{pmatrix}.\n",
"$$\n",
"\n",
"The equilibrium distribution function is defined by\n",
"\n",
"$$\n",
"f^{eq}_{i} = w_{i}\\left( u + \\frac{e_{i,\\alpha} B_{\\alpha}}{c_{s}^{2}} + \\frac{ \\left( c_{s}^{2}C\\delta_{\\alpha\\beta} + \\Lambda_{\\alpha\\beta} - c_{s}^{2}u\\delta_{\\alpha\\beta} \\right) \\left( e_{i,\\alpha}e_{i,\\beta} - c_{s}^{2}\\delta_{\\alpha\\beta} \\right)}{2c_{s}^{4}} \\right),\n",
"$$\n",
"\n",
"where $\\Lambda_{\\alpha\\beta}$ is a correction term defined by\n",
"\n",
"$$\n",
"\\Lambda_{\\alpha\\beta} = \\displaystyle\\int B_{\\alpha}^{'}B_{\\beta}^{'} du = \\displaystyle\\int \\left(\\displaystyle\\frac{\\partial B_{\\alpha}}{\\partial u} \\displaystyle\\frac{\\partial B_{\\beta}}{\\partial u} \\right) du.\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "cc5d6c7b-5352-417b-8403-68f65a467721",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"import warnings\n",
"warnings.filterwarnings(\"ignore\")\n",
"from pylab import *\n",
"from __future__ import division\n",
"from sympy import *\n",
"import numpy as np\n",
"from sympy import S, collect, expand, factor, Wild\n",
"from sympy import fraction, Rational, Symbol\n",
"from sympy import symbols, sqrt, Rational\n",
"import sympy as sp\n",
"from IPython.display import display, Math, Latex\n",
"#-------------------------------------------------Símbolos----------------------------------------------\n",
"omega, u, B, w, C, Lamb = symbols('omega, u, B_{\\\\alpha}, w, C, \\\\Lambda_{\\\\alpha\\\\beta}')\n",
"wi, cx, cy, cs = symbols('w_{i} c_{x} c_{y} c_{s}')\n",
"fi, f0, f1, f2 = symbols('f_{i} f_{0} f_{1} f_{2}')\n",
"#-------------------------------------------------Funções----------------------------------------------\n",
"feq = Function('feq')(wi, cx, cy)\n",
"fneq = Function('fneq')(wi, cx, cy)\n",
"f = Function('f')(fi)\n",
"#----------------------------------------------Lattice-D2Q9---Variáveis----------------------------------------------\n",
"fi=np.array([f0,f1,f2])\n",
"w0=Rational(4,6);w1=Rational(1,6)\n",
"wi=np.array([w0,w1,w1])\n",
"cx=np.array([0,1,-1])\n",
"as2=Rational(3)\n",
"cs2=1/as2\n",
"#-------------------------------------------------Calc.Func------------------------------------------------\n",
"f= fi\n",
"feq=wi*(u + cx*B/cs2 + ((C*cs2 + Lamb -u*cs2)*(cx*cx-cs2))/(2*cs2**2))\n",
"feq2=wi*(u + cx*B/cs2 + (u*C**2*(cx*cx-cs2))/(2*cs2**2))\n",
"feq3=wi*(u + cx*C*u/cs2 )\n",
"feq4=wi*(u + cx*u*C/cs2 + (u*C**2*(cx*cx-cs2))/(2*cs2**2))"
]
},
{
"cell_type": "markdown",
"id": "d58505e6-a477-4760-9d47-e13b5ea16b6a",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": []
},
"source": [
"The different orders of equilibrium moments can be recovered by:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "41bb608a-23b9-4311-8535-de44d14a0310",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\underbrace{\\sum_{i=0} f_{i}^{eq} =\\sum_{i=0} f_{i} }_{\\textrm{Zero-Order Moment}} =u,\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq}e_{i,x} }_{\\textrm{x-First-Order Moment}} =B_{\\alpha},\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} e_{i,x}e_{i,x} }_{\\textrm{xx-Second-Order Moment}} =\\frac{C}{3} + \\Lambda_{\\alpha\\beta},\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} e_{i,x}e_{i,x}e_{i,x} }_{\\textrm{xxx-Third-Order Moment}} =B_{\\alpha}$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\underbrace{\\sum_{i=0} f_{i}^{eq} =\\sum_{i=0} f_{i} }_{\\textrm{Zero-Order Hermite Moment}} =u,\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq}e_{i,x} }_{\\textrm{x-First-Order Hermite Moment}} =B_{\\alpha},\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} \\left(e_{i,x}e_{i,x} - \\frac{1}{c_{s}^{2}}\\right) }_{\\textrm{xx-Second-Order Hermite Moment}} =\\frac{C}{3} + \\Lambda_{\\alpha\\beta} - \\frac{u}{3},\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} \\left(e_{i,x}e_{i,x} - \\frac{3}{c_{s}^{2}}\\right) e_{i,x} }_{\\textrm{xxx-Third-Order Hermite Moment}} =0$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"a0=simplify(sum(feq))\n",
"ax=simplify(sum(feq*cx))\n",
"axx=simplify(sum(feq*cx*cx))\n",
"axxx=simplify(sum(feq*cx*cx*cx))\n",
"display(Math(r\"\\underbrace{\\sum_{i=0} f_{i}^{eq} =\\sum_{i=0} f_{i} }_{\\textrm{Zero-Order Moment}} =\" + sp.latex(a0) \n",
" +r\",\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq}e_{i,x} }_{\\textrm{x-First-Order Moment}} =\" + sp.latex(ax)\n",
" +r\",\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} e_{i,x}e_{i,x} }_{\\textrm{xx-Second-Order Moment}} =\" + sp.latex(axx)\n",
" +r\",\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} e_{i,x}e_{i,x}e_{i,x} }_{\\textrm{xxx-Third-Order Moment}} =\" + sp.latex(axxx) ))\n",
"\n",
"aH0=simplify(sum(feq))\n",
"aHx=simplify(sum(feq*cx))\n",
"aHxx=simplify(sum(feq*(cx*cx-cs2)))\n",
"aHxxx=simplify(sum(feq*(cx*cx-3*cs2)*cx))\n",
"display(Math(r\"\\underbrace{\\sum_{i=0} f_{i}^{eq} =\\sum_{i=0} f_{i} }_{\\textrm{Zero-Order Hermite Moment}} =\" + sp.latex(aH0) \n",
" +r\",\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq}e_{i,x} }_{\\textrm{x-First-Order Hermite Moment}} =\" + sp.latex(aHx)\n",
" +r\",\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} \\left(e_{i,x}e_{i,x} - \\frac{1}{c_{s}^{2}}\\right) }_{\\textrm{xx-Second-Order Hermite Moment}} =\" + sp.latex(aHxx)\n",
" +r\",\\quad \\quad \\underbrace{\\sum_{i=0} f_{i}^{eq} \\left(e_{i,x}e_{i,x} - \\frac{3}{c_{s}^{2}}\\right) e_{i,x} }_{\\textrm{xxx-Third-Order Hermite Moment}} =\" + sp.latex(aHxxx) ))"
]
},
{
"cell_type": "markdown",
"id": "3d9b571d-73ec-425b-be09-0803e063c767",
"metadata": {},
"source": [
"## FDM Interpretation in MRT Framework\n",
"\n",
"Reinterpreting the Eq. {eq}`MRT-LB-conv-df-Eq` with a shift in distribution function direction:\n",
"\n",
"$$\n",
"f_{i,x}^{t+1} = f_{i,x-e_{i}}^{t} - (\\textbf{M}^{-1}\\textbf{S}\\textbf{M})_{ik}\\left( f_{k,x-e_{i}}^{t} - f_{k,x-e_{i}}^{eq,t} \\right) = (\\textbf{M}^{-1} (\\textbf{I} -\\textbf{S}) \\textbf{M})_{ik} f_{k,x-e_{i}}^{t} + (\\textbf{M}^{-1}\\textbf{S}\\textbf{M})_{ik} f_{k,x-e_{i}}^{eq,t} , \n",
"$$\n",
"\n",
"where $f_i( x_{\\alpha} + e_{i,\\alpha} \\delta t, t+\\delta t)=f_{i,x+c_{i}}^{t+1}$ for simplificate the notation and $\\textbf{I}$ is identity matrix. Defining the shift operator $T_{\\Delta x}^{c_{i}}$, where $f_{k,x-e_{i}}^{t}= T_{\\Delta x}^{c_{i}}[f_{k,x}^{t}]$, we can rewrite the above equation inside of the moments space:\n",
"\n",
"$$\n",
"\\textbf{m}_{k,x}^{t+1} = \\textbf{P}\\textbf{m}_{k,x}^{t} + \\textbf{Q}\\textbf{m}_{k,x}^{eq,t}, \n",
"$$(preM-RT-LB-conv-df-Eq)\n",
"\n",
"where $\\textbf{m}_{k,x}^{t}=M_{ik}f_{i,x}^{t}$, $k$ is the index of moments in the space moment, $\\textbf{T}:= \\textbf{M} (diag\\left( T_{\\Delta x}^{c_{0}}, T_{\\Delta x}^{c_{1}}, T_{\\Delta x}^{c_{2}} \\right)) \\textbf{M}^{-1}$, $\\textbf{P}:=\\textbf{T}(\\textbf{I}-\\textbf{S})$, and , $\\textbf{Q}:=\\textbf{T}\\textbf{S}$. The matrices are given by:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "f16da87c-1f91-4105-bd46-7f1191587878",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"import warnings\n",
"warnings.filterwarnings(\"ignore\")\n",
"from pylab import *\n",
"from sympy import *\n",
"import numpy as np\n",
"\n",
"fic, f0c, f1c, f2c = symbols(['f_{i\\,x+c_{i}}^t','f_{0\\,x+c_{i}}^t','f_{1\\,x+c_{i}}^t','f_{2\\,x+c_{i}}^t'])\n",
"f0tp1, f1tp1, f2tp1 = symbols(['f_{0\\,x}^{t+1}','f_{1\\,x}^{t+1}','f_{2\\,x}^{t+1}'])\n",
"f0, f1, f2 = symbols(['f_{0\\,x}^t','f_{1\\,x}^t','f_{2\\,x}^t'])\n",
"f0m1, f1m1, f2m1, f0p1, f1p1, f2p1 = symbols(['f_{0\\,x-1}^t','f_{1\\,x-1}^t','f_{2\\,x-1}^t','f_{0\\,x+1}^t','f_{1\\,x+1}^t','f_{2\\,x+1}^t'])\n",
"f0m1t1, f1m1t1, f2m1t1, f0p1t1, f1p1t1, f2p1t1 = symbols(['f_{0\\,x-1}^{t-1}','f_{1\\,x-1}^{t-1}','f_{2\\,x-1}^{t-1}','f_{0\\,x+1}^{t-1}','f_{1\\,x+1}^{t-1}','f_{2\\,x+1}^{t-1}'])\n",
"f0t1 = symbols('f_{0\\,x}^{t-1}')\n",
"w0, w1, w2 = symbols(['w_0','w_1','w_2'])\n",
"s0, s1, s2 = symbols(['s_0','s_1','s_2'])\n",
"phi, phic, phim1, phip1 = symbols(['\\\\phi_{x}^{t}','\\\\phi_{x-c_{i}}^{t}','\\\\phi_{x-1}^{t}','\\\\phi_{x+1}^{t}'])\n",
"phitp1, phit1, phit2= symbols(['\\\\phi_{x}^{t+1}','\\\\phi_{x}^{t-1}','\\\\phi_{x}^{t-2}'])\n",
"phim1t1,phip1t1= symbols(['\\\\phi_{x-1}^{t-1}','\\\\phi_{x+1}^{t-1}'])\n",
"Tc0, Tc1, Tc2= symbols([r'T_{\\Delta_{x}}^{c_{0}}','T_{\\Delta_{x}}^{c_{1}}','T_{\\Delta_{x}}^{c_{2}}'])\n",
"cx=np.array([0,1,-1])\n",
"wi=np.array([w0,w1,w2])\n",
"wiM=Matrix([w0,w1,w2])\n",
"fi=np.array([f0,f1,f2])\n",
"fM = Matrix([f0,f1,f2])\n",
"ficM = Matrix([f0c,f1c,f2c])\n",
"I=eye(3)\n",
"Tc = Matrix([I[0,:]*Tc0,I[1,:]*Tc1,I[2,:]*Tc2])\n",
"SM=Matrix([I[0,:]*s0,I[1,:]*s1,I[2,:]*s2])\n",
"feq=phi*wi\n",
"feqM=Matrix([feq[0],feq[1],feq[2]])\n",
"feqc=wiM*phic\n",
"m0=np.array([1,1,1])\n",
"m1=cx\n",
"m2=3*cx*cx - 2\n",
"M=np.array([m0,m1,m2])\n",
"MM=Matrix([m0,m1,m2])\n",
"MMinv=MM.inv()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "cc7d06be-17d0-4efd-9398-33cca20fdb1e",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\textbf{T} =\\left[\\begin{matrix}\\frac{T_{\\Delta_{x}}^{c_{0}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{1}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{3} & \\frac{T_{\\Delta_{x}}^{c_{1}}}{2} - \\frac{T_{\\Delta_{x}}^{c_{2}}}{2} & - \\frac{T_{\\Delta_{x}}^{c_{0}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{1}}}{6} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{6}\\\\\\frac{T_{\\Delta_{x}}^{c_{1}}}{3} - \\frac{T_{\\Delta_{x}}^{c_{2}}}{3} & \\frac{T_{\\Delta_{x}}^{c_{1}}}{2} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{2} & \\frac{T_{\\Delta_{x}}^{c_{1}}}{6} - \\frac{T_{\\Delta_{x}}^{c_{2}}}{6}\\\\- \\frac{2 T_{\\Delta_{x}}^{c_{0}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{1}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{3} & \\frac{T_{\\Delta_{x}}^{c_{1}}}{2} - \\frac{T_{\\Delta_{x}}^{c_{2}}}{2} & \\frac{2 T_{\\Delta_{x}}^{c_{0}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{1}}}{6} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{6}\\end{matrix}\\right],$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\textbf{P} =\\left[\\begin{matrix}- \\frac{\\left(s_{0} - 1\\right) \\left(T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{3} & - \\frac{\\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right) \\left(s_{1} - 1\\right)}{2} & - \\frac{\\left(s_{2} - 1\\right) \\left(- 2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{6}\\\\- \\frac{\\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right) \\left(s_{0} - 1\\right)}{3} & - \\frac{\\left(T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right) \\left(s_{1} - 1\\right)}{2} & - \\frac{\\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right) \\left(s_{2} - 1\\right)}{6}\\\\- \\frac{\\left(s_{0} - 1\\right) \\left(- 2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{3} & - \\frac{\\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right) \\left(s_{1} - 1\\right)}{2} & - \\frac{\\left(s_{2} - 1\\right) \\left(4 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{6}\\end{matrix}\\right],$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\textbf{Q} =\\left[\\begin{matrix}\\frac{s_{0} \\left(T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{3} & \\frac{s_{1} \\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right)}{2} & \\frac{s_{2} \\left(- 2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{6}\\\\\\frac{s_{0} \\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right)}{3} & \\frac{s_{1} \\left(T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{2} & \\frac{s_{2} \\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right)}{6}\\\\\\frac{s_{0} \\left(- 2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{3} & \\frac{s_{1} \\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right)}{2} & \\frac{s_{2} \\left(4 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{6}\\end{matrix}\\right].$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"TM=sp.simplify(MM*Tc*MMinv)\n",
"P=sp.simplify(TM*(I-SM))\n",
"Q=sp.simplify(TM*SM)\n",
"\n",
"display(Math(r\"\\textbf{T} =\" + sp.latex(TM) +\",\" ))\n",
"display(Math(r\"\\textbf{P} =\" + sp.latex(P) +\",\" ))\n",
"display(Math(r\"\\textbf{Q} =\" + sp.latex(Q) +\".\" ))"
]
},
{
"cell_type": "markdown",
"id": "db0374d5-3549-490d-af27-9d498dacfe0f",
"metadata": {},
"source": [
"By the Eq. {eq}`preM-RT-LB-conv-df-Eq` can be shifted in time, we recursively over $n$ timestep replace the shifted time function in it:\n",
"\n",
"$$\n",
"\\underbrace{ \\textbf{m}_{k,x}^{t+1} = \\textbf{P}\\textbf{m}_{k,x}^{t} + \\textbf{Q}\\textbf{m}_{k,x}^{eq,t} \\quad \\rightarrow \\quad \\textbf{m}_{k,x}^{t} = \\textbf{P}\\textbf{m}_{k,x}^{t-1} + \\textbf{Q}\\textbf{m}_{k,x}^{eq,t-1} }_{\\textrm{Time Shift}} \\qquad \\textrm{Recursive Substitution $n$ times:} \\qquad \\textbf{m}_{k,x}^{t+1} = \\textbf{P}^{n}\\textbf{m}_{k,x}^{t-n+1} + \\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,t-l}.\n",
"$$(preM-RT-LB-conv-df-Eq-s2)"
]
},
{
"cell_type": "markdown",
"id": "83fc2fe8-8503-4926-b4d4-e540d9f93c51",
"metadata": {
"editable": true,
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"slide_type": ""
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"tags": []
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"source": [
"The next steps follow in orde to employ the Calay-Hamilton theorem {cite:p}`bellotti2022finite`. In htis way, we need to find the characteristical polinomial $\\mathcal{X}_{P}$ of the matrix $\\textbf{P}$ (the definition of characteristical polinomial are presented in toogle below), given by:\n",
"\n",
"```{toggle}\n",
"The characteristic polynomial of a square matrix $(P \\in \\mathbb{R}^{n \\times n})$ (or $(\\mathbb{C}^{n \\times n}))$ is defined as:\n",
"\n",
"$$\n",
"\\boxed{\\mathcal{X}_{P}(X) = \\det(X I - P)} \\qquad \\rightarrow \\qquad \\mathcal{X}_{P}(X) = \\sum_{i=0}^{n} X^{i}\\gamma^{i} \n",
"$$\n",
"\n",
"where $\\gamma_{i}$ are the polynomial coefficients.\n",
"\n",
"```\n",
"\n",
"$$\n",
"\\mathcal{X}_{P} = \\gamma_{3}X^{3} + \\gamma_{2}X^{2} + \\gamma_{1}X^{1} + \\gamma_{0},\n",
"$$\n",
"\n",
"where all the polynomial coefficients $\\gamma_{i}$ are shifted by $T_{\\Delta x}^{c_{0}}$:"
]
},
{
"cell_type": "code",
"execution_count": 113,
"id": "6990db06-dc71-4536-8a79-39b4ba4ffd68",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
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},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\gamma_{3} =T_{\\Delta_{x}}^{c_{0}},$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\gamma_{2} =- T_{\\Delta_{x}}^{c_{0}} - T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}} + s_{0} \\left(\\frac{T_{\\Delta_{x}}^{c_{0}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{1}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{3}\\right) + s_{1} \\left(\\frac{T_{\\Delta_{x}}^{c_{1}}}{2} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{2}\\right) + s_{2} \\left(\\frac{2 T_{\\Delta_{x}}^{c_{0}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{1}}}{6} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{6}\\right),$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\gamma_{1} =T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}} + \\frac{s_{1} s_{2} \\left(T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{3} - \\frac{s_{1} \\left(2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{2} - \\frac{s_{2} \\left(2 T_{\\Delta_{x}}^{c_{0}} + 5 T_{\\Delta_{x}}^{c_{1}} + 5 T_{\\Delta_{x}}^{c_{2}}\\right)}{6},$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\gamma_{0} =- T_{\\Delta_{x}}^{c_{0}} \\left(s_{1} - 1\\right) \\left(s_{2} - 1\\right).$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from sympy import Matrix, symbols\n",
"X = symbols('X')\n",
"cp = P.charpoly(X)\n",
"# cp.as_expr() \n",
"gamma3=Tc0\n",
"gamma2=sp.collect(cp.all_coeffs()[1],[s2,s1,s0])\n",
"gamma1=sp.simplify(sp.collect(cp.all_coeffs()[2],[s2*s1, s2, s1,s0]).subs(s0,0).subs(Tc1*Tc2,Tc0).subs(Tc0,1)).subs(2,2*Tc0).subs(1,Tc0)\n",
"gamma0=sp.factor(sp.collect(cp.all_coeffs()[3],[s2*s1, s2, s1,s0]).subs(s0,0).subs(Tc1*Tc2,Tc0).subs(Tc0,1))*Tc0\n",
"\n",
"display(Math(r\"\\gamma_{3} =\" + sp.latex(gamma3) +\",\" ))\n",
"display(Math(r\"\\gamma_{2} =\" + sp.latex(gamma2) +\",\" ))\n",
"display(Math(r\"\\gamma_{1} =\" + sp.latex(gamma1) +\",\" ))\n",
"display(Math(r\"\\gamma_{0} =\" + sp.latex(gamma0) +\".\" ))"
]
},
{
"cell_type": "markdown",
"id": "36052f1b-17f1-48b2-8204-7efbb864e6a1",
"metadata": {},
"source": [
"To reach the above coeficient results, were considered: $T_{\\Delta x}^{c_{1}}T_{\\Delta x}^{c_{2}}=T_{\\Delta x}^{c_{0}}$; $T_{\\Delta x}^{c_{1}}T_{\\Delta x}^{c_{0}}=T_{\\Delta x}^{c_{1}}$; $T_{\\Delta x}^{c_{2}}T_{\\Delta x}^{c_{0}}=T_{\\Delta x}^{c_{2}}$.Defined the characteristical polynomial, we multiply both sides of Eq. {eq}`preM-RT-LB-conv-df-Eq-s2` by $\\sum_{n=0}^{q}\\gamma_{n}$ (where $q$ is the number of lattice velocities) and perform a time shift by change the time variable $t$ to $\\tilde{t}+n=t+1$:\n",
"\n",
"$$\n",
"\\begin{array}{c}\n",
"\\textbf{m}_{k,x}^{t+1} = \\textbf{P}^{n}\\textbf{m}_{k,x}^{t-n+1} + \\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,t-l} \\qquad \\rightarrow \\qquad \\textbf{m}_{k,x}^{\\tilde{t}+n} = \\textbf{P}^{n}\\textbf{m}_{k,x}^{\\tilde{t}} + \\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,\\tilde{t}+n-1-l} \\qquad \\rightarrow \\\\\n",
"\\displaystyle\\sum_{n=0}^{q}\\gamma_{n} \\textbf{m}_{k,x}^{\\tilde{t}+n} = \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\textbf{P}^{n}\\textbf{m}_{k,x}^{\\tilde{t}} + \\displaystyle\\sum_{n=0}^{q}\\gamma_{n} \\left(\\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,\\tilde{t}+n-1-l}\\right),\n",
"\\end{array}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "75720ea7-89ee-4f34-86f0-6cced49a95fe",
"metadata": {},
"source": [
"by applying the Calay-Hamilton theorem, that imlpie $\\sum_{n=0}^{q}\\gamma_{n}\\textbf{P}^{n}=0$:"
]
},
{
"cell_type": "markdown",
"id": "b42deace-e799-406f-8d4d-6eb675b7048f",
"metadata": {},
"source": [
"$$\n",
"\\displaystyle\\sum_{n=0}^{q}\\gamma_{n} \\textbf{m}_{k,x}^{\\tilde{t}+n} = \\underbrace{\\left(\\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\textbf{P}^{n}\\right)}_{=0}\\textbf{m}_{k,x}^{\\tilde{t}} + \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\left( \\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,\\tilde{t}+n-1-l} \\right) \\qquad \\rightarrow \\qquad \\displaystyle\\sum_{n=0}^{q}\\gamma_{n} \\textbf{m}_{k,x}^{\\tilde{t}+n} = \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\left(\\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,\\tilde{t}+n-1-l}\\right),\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "afb3af3b-ce1e-4b16-99a5-6def76a1d925",
"metadata": {},
"source": [
"applying a second time shift by change tha variable back to $\\tilde{t}+q=t+1$ and expanding the left hand-side term to isolate the time $t+1$:"
]
},
{
"cell_type": "markdown",
"id": "2a27ef68-3634-4a6b-911c-6c36be8530dd",
"metadata": {},
"source": [
"$$\n",
"\\begin{array}{c}\n",
"\\displaystyle\\sum_{n=0}^{q}\\gamma_{n} \\textbf{m}_{k,x}^{\\tilde{t}+n} = \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\left(\\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,\\tilde{t}+n-1-l}\\right) \\qquad \\rightarrow \\qquad \\displaystyle\\sum_{n=0}^{q}\\gamma_{n} \\textbf{m}_{k,x}^{t+1-q+n} = \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\left(\\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,t-q+n-l}\\right) \\qquad \\rightarrow \\\\\n",
"\\displaystyle\\sum_{n=0}^{q}\\gamma_{n} \\textbf{m}_{k,x}^{t+1-q+n} = \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\left(\\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,t-q+n-l}\\right) \\qquad \\rightarrow \\qquad \\gamma_{q} \\textbf{m}_{k,x}^{t+1} = -\\displaystyle\\sum_{n=0}^{q-1}\\gamma_{n} \\textbf{m}_{k,x}^{t+1-q+n} + \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\left(\\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,t-q+n-l}\\right),\n",
"\\end{array}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "c0e8f0c6-05b6-4697-9b4d-ae9df16a3f23",
"metadata": {},
"source": [
"the last sum term can be rewritten by\n",
"\n",
"$$\n",
"\\gamma_{q} \\textbf{m}_{k,x}^{t+1} = -\\displaystyle\\sum_{n=0}^{q-1}\\gamma_{n} \\textbf{m}_{k,x}^{t+1-q+n} + \\displaystyle\\sum_{n=0}^{q}\\gamma_{n}\\left(\\displaystyle\\sum_{l=0}^{n-1}\\textbf{P}^{l}\\textbf{Q}\\textbf{m}_{k,x}^{eq,t-q+n-l}\\right) \\qquad \\rightarrow \\qquad \\gamma_{q} \\textbf{m}_{k,x}^{t+1} = -\\displaystyle\\sum_{n=0}^{q-1}\\gamma_{n} \\textbf{m}_{k,x}^{t+1-q+n} + \\displaystyle\\sum_{n=0}^{q-1}\\left(\\displaystyle\\sum_{l=0}^{n}\\gamma_{q+l-n}\\textbf{P}^{l}\\right)\\textbf{Q}\\textbf{m}_{k,x}^{eq,t-n}.\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 114,
"id": "2de29dc9-4153-4247-a6f7-8e1b66af5ea9",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle m^{t+1}_{k,x} {\\gamma}_{3} = P^{2} Q m^{eq,t-2}_{k,x} {\\gamma}_{3} + P Q m^{eq,t-1}_{k,x} {\\gamma}_{3} + P Q m^{eq,t-2}_{k,x} {\\gamma}_{2} + Q m^{eq,t-1}_{k,x} {\\gamma}_{2} + Q m^{eq,t-2}_{k,x} {\\gamma}_{1} + Q m^{eq,t}_{k,x} {\\gamma}_{3} - m^{t-1}_{k,x} {\\gamma}_{1} - m^{t-2}_{k,x} {\\gamma}_{0} - m^{t}_{k,x} {\\gamma}_{2}$"
],
"text/plain": [
"Eq(m^{t+1}_{k,x}*gamma[3], P**2*Q*m^{eq,t-2}_{k,x}*gamma[3] + P*Q*m^{eq,t-1}_{k,x}*gamma[3] + P*Q*m^{eq,t-2}_{k,x}*gamma[2] + Q*m^{eq,t-1}_{k,x}*gamma[2] + Q*m^{eq,t-2}_{k,x}*gamma[1] + Q*m^{eq,t}_{k,x}*gamma[3] - m^{t-1}_{k,x}*gamma[1] - m^{t-2}_{k,x}*gamma[0] - m^{t}_{k,x}*gamma[2])"
]
},
"execution_count": 114,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import sympy as sp\n",
"# ---------------- indices / parameters ----------------\n",
"q = sp.Symbol('q', integer=True, positive=True)\n",
"n, l = sp.symbols('n l', integer=True, nonnegative=True)\n",
"m1s = sp.Symbol('m^{t+1}_{k,x}')\n",
"m0s = sp.Symbol('m^{t}_{k,x}')\n",
"mm1s = sp.Symbol('m^{t-1}_{k,x}')\n",
"mm2s = sp.Symbol('m^{t-2}_{k,x}')\n",
"meq0s = sp.Symbol('m^{eq,t}_{k,x}')\n",
"meqm1s = sp.Symbol('m^{eq,t-1}_{k,x}')\n",
"meqm2s = sp.Symbol('m^{eq,t-2}_{k,x}')\n",
"\n",
"# gamma_j as an indexed coefficient sequence\n",
"gammas = sp.IndexedBase('gamma') # gamma[j] means γ_j\n",
"\n",
"# Abstract time-shifted moments:\n",
"# m(s) := m_{k,x}^{t+s}\n",
"# meq(s) := m_{k,x}^{eq, t+s}\n",
"ms = sp.Function('m')\n",
"meqs = sp.Function('m_eq')\n",
"\n",
"# Operators/matrices (leave as Symbols unless you need matrix algebra)\n",
"Ps = sp.Symbol('P')\n",
"Qs = sp.Symbol('Q')\n",
"\n",
"# ---------------- build the equation ----------------\n",
"lhs = gammas[q] * ms(1) # γ_q * m^{t+1}\n",
"\n",
"rhs1 = -sp.summation(gammas[n] * ms(1 - q + n), (n, 0, q-1))\n",
"\n",
"inner = sp.summation(gammas[q + l - n] * Ps**l, (l, 0, n))\n",
"rhs2 = sp.summation(inner * Qs * meqs(-n), (n, 0, q-1))\n",
"\n",
"# eq = sp.Eq(lhs, rhs1 + rhs2)\n",
"eq = rhs1 + rhs2\n",
"eq1 = lhs.subs({ms(1):m1s,gammas[q]:gammas[3]})\n",
"\n",
"# ---------------- expand for a concrete q ----------------\n",
"# SymPy can only 'doit' finite sums once q is a number.\n",
"q_val = 3 # <-- change to 2,3,4,... as needed\n",
"eq_expanded = sp.expand(eq.subs(q, q_val).doit())\n",
"\n",
"# print(f\"\\nExpanded for q={q_val}:\")\n",
"# display(eq_expanded)\n",
"\n",
"eqsubs=eq_expanded.subs({ms(1):m1s,ms(0):m0s,ms(-1):mm1s,ms(-2):mm2s,meqs(0):meq0s,meqs(0):meq0s,meqs(-1):meqm1s,meqs(-2):meqm2s})\n",
"sp.Eq(eq1, eqsubs)"
]
},
{
"cell_type": "markdown",
"id": "85a1a931-51b7-4618-bc4d-6415ec498736",
"metadata": {},
"source": [
"Considering the solution for the moment $m^{t+1}_{0,x}$, the above equation is rewritten to:\n",
"\n",
"$$\n",
"m^{t+1}_{0,x} \\gamma_{3} = \\left[P^{2} Q m^{eq,t-2}_{k,x} \\gamma_{3}\\right]_{k=0} + \\left[P Q m^{eq,t-1}_{k,x} \\gamma_{3}\\right]_{k=0} + \\left[P Q m^{eq,t-2}_{k,x} \\gamma_{2}\\right]_{k=0} + \\left[Q m^{eq,t-1}_{k,x} \\gamma_{2}\\right]_{k=0} + \\left[Q m^{eq,t-2}_{k,x} \\gamma_{1}\\right]_{k=0} + \\left[Q m^{eq,t}_{k,x} \\gamma_{3}\\right]_{k=0} - m^{t-1}_{0,x} \\gamma_{1} - m^{t-2}_{0,x} \\gamma_{0} - m^{t}_{0,x} \\gamma_{2}\n",
"$$(Momentum-exp-term-convec-diff)"
]
},
{
"cell_type": "code",
"execution_count": 115,
"id": "402738ac-a6bb-4c9a-835e-d82884308442",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"import sympy as sp\n",
"# ---------------- indices / parameters ----------------\n",
"meq0k11s = sp.Symbol('m^{eq,t+1}_{0,x}')\n",
"meq0k0s = sp.Symbol('m^{eq,t}_{0,x}')\n",
"meq1k0s = sp.Symbol('m^{eq,t}_{1,x}')\n",
"meq2k0s = sp.Symbol('m^{eq,t}_{2,x}')\n",
"MeqM0s=Matrix([meq0k0s,meq1k0s,meq2k0s])\n",
"meq0k1s = sp.Symbol('m^{eq,t-1}_{0,x}')\n",
"meq1k1s = sp.Symbol('m^{eq,t-1}_{1,x}')\n",
"meq2k1s = sp.Symbol('m^{eq,t-1}_{2,x}')\n",
"MeqM1s=Matrix([meq0k1s,meq1k1s,meq2k1s])\n",
"meq0k2s = sp.Symbol('m^{eq,t-2}_{0,x}')\n",
"meq1k2s = sp.Symbol('m^{eq,t-2}_{1,x}')\n",
"meq2k2s = sp.Symbol('m^{eq,t-2}_{2,x}')\n",
"MeqM2s=Matrix([meq0k2s,meq1k2s,meq2k2s])\n",
"m0k11s = sp.Symbol('m^{t+1}_{0,x}')\n",
"m0k0s = sp.Symbol('m^{t}_{0,x}')\n",
"m0k1s = sp.Symbol('m^{t-1}_{0,x}')\n",
"m0k2s = sp.Symbol('m^{t-2}_{0,x}')"
]
},
{
"cell_type": "code",
"execution_count": 116,
"id": "2baa68fd-c662-48c2-90aa-51489bbe64d6",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[ P Q m^{eq,t-1}_{k,x} \\gamma_{3} \\right]_{k=0} + \\left[ Q m^{eq,t-1}_{k,x} \\gamma_{2} \\right]_{k=0} =m^{eq,t-1}_{1,x} \\left(\\frac{T_{\\Delta_{x}}^{c_{1}} s_{1} s_{2}}{2} - \\frac{T_{\\Delta_{x}}^{c_{1}} s_{1}}{2} - \\frac{T_{\\Delta_{x}}^{c_{2}} s_{1} s_{2}}{2} + \\frac{T_{\\Delta_{x}}^{c_{2}} s_{1}}{2}\\right) + m^{eq,t-1}_{2,x} \\left(\\frac{T_{\\Delta_{x}}^{c_{0}} s_{1} s_{2}}{3} - \\frac{T_{\\Delta_{x}}^{c_{0}} s_{2}}{3} - \\frac{T_{\\Delta_{x}}^{c_{1}} s_{1} s_{2}}{6} + \\frac{T_{\\Delta_{x}}^{c_{1}} s_{2}}{6} - \\frac{T_{\\Delta_{x}}^{c_{2}} s_{1} s_{2}}{6} + \\frac{T_{\\Delta_{x}}^{c_{2}} s_{2}}{6}\\right),$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"termmeqm1=collect(simplify(expand(Q*MeqM1s*gamma2 + P*Q*MeqM1s))[0].subs(Tc0*Tc1,Tc1).subs(Tc0*Tc2,Tc2).subs(Tc1*Tc2,Tc0).subs(s0,0),[meq0k1s,meq1k1s,meq2k1s])\n",
"termmeqm1\n",
"display(Math(r\"\\left[ P Q m^{eq,t-1}_{k,x} \\gamma_{3} \\right]_{k=0} + \\left[ Q m^{eq,t-1}_{k,x} \\gamma_{2} \\right]_{k=0} =\" + sp.latex(termmeqm1) +\",\" ))"
]
},
{
"cell_type": "code",
"execution_count": 117,
"id": "92a96029-9bf9-49ad-af69-7c0129e8949e",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[ P^{2} Q m^{eq,t-2}_{k,x} \\gamma_{3} \\right]_{k=0} + \\left[ PQ m^{eq,t-2}_{k,x} \\gamma_{2} \\right]_{k=0} + \\left[ Q m^{eq,t-2}_{k,x} \\gamma_{1} \\right]_{k=0} =0,$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# termmeqm2 = expand((P*Q)*MeqMs*gamma2 + Q*MeqMs*gamma1)[0]\n",
"termmeqm2 = expand( ((P*P)*Q)*MeqM2s*gamma3 + (P*Q)*MeqM2s*gamma2 + Q*MeqM2s*gamma1)[0].subs(Tc1*Tc2,Tc0).subs(Tc0,1).subs(s0,0)\n",
"display(Math(r\"\\left[ P^{2} Q m^{eq,t-2}_{k,x} \\gamma_{3} \\right]_{k=0} + \\left[ PQ m^{eq,t-2}_{k,x} \\gamma_{2} \\right]_{k=0} + \\left[ Q m^{eq,t-2}_{k,x} \\gamma_{1} \\right]_{k=0} =\" + sp.latex(termmeqm2) +\",\" ))"
]
},
{
"cell_type": "code",
"execution_count": 118,
"id": "1eb1a90a-955b-4fdc-9fad-c9a5fdc1164e",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[Q m^{eq,t}_{k,x} \\gamma_{3} \\right]_{k=0} =\\frac{m^{eq,t}_{1,x} s_{1} \\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right)}{2} + \\frac{m^{eq,t}_{2,x} s_{2} \\left(- 2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{6}.$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"termmeqm3 = (Q*MeqM0s)[0].subs(s0,0)\n",
"# sp.Eq(Qs*meq0s*gammas[3], termmeqm3)\n",
"display(Math(r\"\\left[Q m^{eq,t}_{k,x} \\gamma_{3} \\right]_{k=0} =\" + sp.latex(termmeqm3) +\".\" ))"
]
},
{
"cell_type": "markdown",
"id": "2a751b2a-85d6-4a72-8963-69e729491f33",
"metadata": {},
"source": [
"Replacing the terms in the Eq. {eq}`Momentum-exp-term-convec-diff`:"
]
},
{
"cell_type": "code",
"execution_count": 119,
"id": "e1692423-de10-40a4-829d-c0cfe7af0739",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle T_{\\Delta_{x}}^{c_{0}} m^{t+1}_{0,x} = T_{\\Delta_{x}}^{c_{0}} m^{t-2}_{0,x} \\left(s_{1} - 1\\right) \\left(s_{2} - 1\\right) + \\frac{m^{eq,t-1}_{1,x} s_{1} \\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right) \\left(s_{2} - 1\\right)}{2} + \\frac{m^{eq,t-1}_{2,x} s_{2} \\left(s_{1} - 1\\right) \\left(2 T_{\\Delta_{x}}^{c_{0}} - T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right)}{6} + \\frac{m^{eq,t}_{1,x} s_{1} \\left(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}}\\right)}{2} + \\frac{m^{eq,t}_{2,x} s_{2} \\left(- 2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{6} - m^{t-1}_{0,x} \\left(T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}} + \\frac{s_{1} s_{2} \\left(T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{3} - \\frac{s_{1} \\left(2 T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}}\\right)}{2} - \\frac{s_{2} \\left(2 T_{\\Delta_{x}}^{c_{0}} + 5 T_{\\Delta_{x}}^{c_{1}} + 5 T_{\\Delta_{x}}^{c_{2}}\\right)}{6}\\right) - m^{t}_{0,x} \\left(- T_{\\Delta_{x}}^{c_{0}} - T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}} + s_{1} \\left(\\frac{T_{\\Delta_{x}}^{c_{1}}}{2} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{2}\\right) + s_{2} \\left(\\frac{2 T_{\\Delta_{x}}^{c_{0}}}{3} + \\frac{T_{\\Delta_{x}}^{c_{1}}}{6} + \\frac{T_{\\Delta_{x}}^{c_{2}}}{6}\\right)\\right)$"
],
"text/plain": [
"Eq(T_{\\Delta_{x}}^{c_{0}}*m^{t+1}_{0,x}, T_{\\Delta_{x}}^{c_{0}}*m^{t-2}_{0,x}*(s_1 - 1)*(s_2 - 1) + m^{eq,t-1}_{1,x}*s_1*(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}})*(s_2 - 1)/2 + m^{eq,t-1}_{2,x}*s_2*(s_1 - 1)*(2*T_{\\Delta_{x}}^{c_{0}} - T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}})/6 + m^{eq,t}_{1,x}*s_1*(T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}})/2 + m^{eq,t}_{2,x}*s_2*(-2*T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}})/6 - m^{t-1}_{0,x}*(T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}} + s_1*s_2*(T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}})/3 - s_1*(2*T_{\\Delta_{x}}^{c_{0}} + T_{\\Delta_{x}}^{c_{1}} + T_{\\Delta_{x}}^{c_{2}})/2 - s_2*(2*T_{\\Delta_{x}}^{c_{0}} + 5*T_{\\Delta_{x}}^{c_{1}} + 5*T_{\\Delta_{x}}^{c_{2}})/6) - m^{t}_{0,x}*(-T_{\\Delta_{x}}^{c_{0}} - T_{\\Delta_{x}}^{c_{1}} - T_{\\Delta_{x}}^{c_{2}} + s_1*(T_{\\Delta_{x}}^{c_{1}}/2 + T_{\\Delta_{x}}^{c_{2}}/2) + s_2*(2*T_{\\Delta_{x}}^{c_{0}}/3 + T_{\\Delta_{x}}^{c_{1}}/6 + T_{\\Delta_{x}}^{c_{2}}/6)))"
]
},
"execution_count": 119,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eqsubs2=(eqsubs.subs({meqm2s:0,Ps*Qs*meqm1s*gammas[3] + Qs*meqm1s*gammas[2]:termmeqm1,Qs*meq0s*gammas[3]:termmeqm3})\n",
" .subs({m0s:m0k0s,mm1s:m0k1s,mm2s:m0k2s}).subs({gammas[3]:gamma3,gammas[2]:gamma2,gammas[1]:gamma1,gammas[0]:gamma0}).subs({s0:0})\n",
" .subs({expand((Tc1*s1*s2-Tc1*s1-Tc2*s1*s2+Tc2*s1)/2):factor(expand((Tc1*s1*s2-Tc1*s1-Tc2*s1*s2+Tc2*s1)/2))})\n",
" .subs({expand((2*Tc0*s1*s2-2*Tc0*s2-Tc1*s1*s2+Tc1*s2-Tc2*s1*s2+Tc2*s2)/6):factor(expand((2*Tc0*s1*s2-2*Tc0*s2-Tc1*s1*s2+Tc1*s2-Tc2*s1*s2+Tc2*s2)/6))}) )\n",
"sp.Eq(eq1.subs({m1s:m0k11s,gammas[3]:gamma3}), eqsubs2)"
]
},
{
"cell_type": "code",
"execution_count": 120,
"id": "7e77b925-0ec3-4f1b-a1d8-d5aa6257f7da",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"m0k0sdp1 = sp.Symbol('m^{t}_{0,x+1}')\n",
"m0k0sdm1 = sp.Symbol('m^{t}_{0,x-1}')\n",
"m0k1sdp1 = sp.Symbol('m^{t-1}_{0,x+1}')\n",
"m0k1sdm1 = sp.Symbol('m^{t-1}_{0,x-1}')\n",
"\n",
"meq1k0sdp1 = sp.Symbol('m^{eq,t}_{1,x+1}')\n",
"meq1k0sdm1 = sp.Symbol('m^{eq,t}_{1,x-1}')\n",
"meq2k0sdp1 = sp.Symbol('m^{eq,t}_{2,x+1}')\n",
"meq2k0sdm1 = sp.Symbol('m^{eq,t}_{2,x-1}')\n",
"\n",
"meq1k1sdp1 = sp.Symbol('m^{eq,t-1}_{1,x+1}')\n",
"meq1k1sdm1 = sp.Symbol('m^{eq,t-1}_{1,x-1}')\n",
"meq2k1sdp1 = sp.Symbol('m^{eq,t-1}_{2,x+1}')\n",
"meq2k1sdm1 = sp.Symbol('m^{eq,t-1}_{2,x-1}')"
]
},
{
"cell_type": "code",
"execution_count": 121,
"id": "e9f0abea-b1db-49cc-a7be-ea964274f7a0",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle m^{t+1}_{0,x} = m^{eq,t-1}_{1,x+1} \\left(- \\frac{s_{1} s_{2}}{2} + \\frac{s_{1}}{2}\\right) + m^{eq,t-1}_{1,x-1} \\left(\\frac{s_{1} s_{2}}{2} - \\frac{s_{1}}{2}\\right) + m^{eq,t-1}_{2,x+1} \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) + m^{eq,t-1}_{2,x-1} \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) + m^{eq,t-1}_{2,x} \\left(\\frac{s_{1} s_{2}}{3} - \\frac{s_{2}}{3}\\right) - \\frac{m^{eq,t}_{1,x+1} s_{1}}{2} + \\frac{m^{eq,t}_{1,x-1} s_{1}}{2} + \\frac{m^{eq,t}_{2,x+1} s_{2}}{6} + \\frac{m^{eq,t}_{2,x-1} s_{2}}{6} - \\frac{m^{eq,t}_{2,x} s_{2}}{3} + m^{t-1}_{0,x+1} \\left(- \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} - 1\\right) + m^{t-1}_{0,x-1} \\left(- \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} - 1\\right) + m^{t-1}_{0,x} \\left(- \\frac{s_{1} s_{2}}{3} + s_{1} + \\frac{s_{2}}{3} - 1\\right) + m^{t-2}_{0,x} \\left(s_{1} - 1\\right) \\left(s_{2} - 1\\right) + m^{t}_{0,x+1} \\left(- \\frac{s_{1}}{2} - \\frac{s_{2}}{6} + 1\\right) + m^{t}_{0,x-1} \\left(- \\frac{s_{1}}{2} - \\frac{s_{2}}{6} + 1\\right) + m^{t}_{0,x} \\left(1 - \\frac{2 s_{2}}{3}\\right)$"
],
"text/plain": [
"Eq(m^{t+1}_{0,x}, m^{eq,t-1}_{1,x+1}*(-s_1*s_2/2 + s_1/2) + m^{eq,t-1}_{1,x-1}*(s_1*s_2/2 - s_1/2) + m^{eq,t-1}_{2,x+1}*(-s_1*s_2/6 + s_2/6) + m^{eq,t-1}_{2,x-1}*(-s_1*s_2/6 + s_2/6) + m^{eq,t-1}_{2,x}*(s_1*s_2/3 - s_2/3) - m^{eq,t}_{1,x+1}*s_1/2 + m^{eq,t}_{1,x-1}*s_1/2 + m^{eq,t}_{2,x+1}*s_2/6 + m^{eq,t}_{2,x-1}*s_2/6 - m^{eq,t}_{2,x}*s_2/3 + m^{t-1}_{0,x+1}*(-s_1*s_2/3 + s_1/2 + 5*s_2/6 - 1) + m^{t-1}_{0,x-1}*(-s_1*s_2/3 + s_1/2 + 5*s_2/6 - 1) + m^{t-1}_{0,x}*(-s_1*s_2/3 + s_1 + s_2/3 - 1) + m^{t-2}_{0,x}*(s_1 - 1)*(s_2 - 1) + m^{t}_{0,x+1}*(-s_1/2 - s_2/6 + 1) + m^{t}_{0,x-1}*(-s_1/2 - s_2/6 + 1) + m^{t}_{0,x}*(1 - 2*s_2/3))"
]
},
"execution_count": 121,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# eqsubs3 = (collect(expand(eqsubs2).subs({Tc0:1})\n",
"# .subs({Tc1*m0k0s:m0k0sdp1,Tc2*m0k0s:m0k0sdm1})\n",
"# .subs({Tc1*m0k1s:m0k1sdp1,Tc2*m0k1s:m0k1sdm1})\n",
"# .subs({Tc1*meq1k0s:meq1k0sdp1,Tc2*meq1k0s:meq1k0sdm1})\n",
"# .subs({Tc1*meq2k0s:meq2k0sdp1,Tc2*meq2k0s:meq2k0sdm1})\n",
"# .subs({Tc1*meq1k1s:meq1k1sdp1,Tc2*meq1k1s:meq1k1sdm1})\n",
"# .subs({Tc1*meq2k1s:meq2k1sdp1,Tc2*meq2k1s:meq2k1sdm1})\n",
"# ,[m0k2s, m0k0sdp1,m0k0sdm1,m0k0s, m0k1sdm1,m0k1sdp1,m0k1s, meq1k0sdm1,meq1k0sdp1,meq1k0s, meq2k0sdm1,meq2k0sdp1,meq2k0s,\n",
"# meq1k1sdm1,meq1k1sdp1,meq1k1s, meq2k1sdm1,meq2k1sdp1,meq2k1s])\n",
"# .subs({(s1*s2-s1-s2+1):factor(s1*s2-s1-s2+1)}) \n",
"# )\n",
"eqsubs3 = (collect(expand(eqsubs2).subs({Tc0:1})\n",
" .subs({Tc2*m0k0s:m0k0sdp1,Tc1*m0k0s:m0k0sdm1})\n",
" .subs({Tc2*m0k1s:m0k1sdp1,Tc1*m0k1s:m0k1sdm1})\n",
" .subs({Tc2*meq1k0s:meq1k0sdp1,Tc1*meq1k0s:meq1k0sdm1})\n",
" .subs({Tc2*meq2k0s:meq2k0sdp1,Tc1*meq2k0s:meq2k0sdm1})\n",
" .subs({Tc2*meq1k1s:meq1k1sdp1,Tc1*meq1k1s:meq1k1sdm1})\n",
" .subs({Tc2*meq2k1s:meq2k1sdp1,Tc1*meq2k1s:meq2k1sdm1})\n",
" ,[m0k2s, m0k0sdp1,m0k0sdm1,m0k0s, m0k1sdm1,m0k1sdp1,m0k1s, meq1k0sdm1,meq1k0sdp1,meq1k0s, meq2k0sdm1,meq2k0sdp1,meq2k0s,\n",
" meq1k1sdm1,meq1k1sdp1,meq1k1s, meq2k1sdm1,meq2k1sdp1,meq2k1s])\n",
" .subs({(s1*s2-s1-s2+1):factor(s1*s2-s1-s2+1)}) \n",
" )\n",
"sp.Eq(eq1.subs({m1s:m0k11s,gammas[3]:1}), eqsubs3)"
]
},
{
"cell_type": "markdown",
"id": "478c3f9d-4634-43d5-9971-27446d56e645",
"metadata": {},
"source": [
"Replacing the moments:\n",
"\n",
"$$\n",
"m_{0,x}^{t}=u_{x}^{t}, \\qquad m_{0,x}^{eq,t}=u_{x}^{t}, \\qquad m_{1,x}^{eq,t}=B_{x}^{t}, \\qquad \\textrm{and} \\qquad m_{1,x}^{eq,t}=C_{x}^{t} + 3\\Lambda_{x}^{t} - 2u_{x}^{t},\n",
"$$\n",
"\n",
"in the above equation:"
]
},
{
"cell_type": "code",
"execution_count": 122,
"id": "70a77963-63e7-4e68-98be-a9569f572a65",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"u11s = sp.Symbol('u^{t+1}_{x}')\n",
"u0s = sp.Symbol('u^{t}_{x}')\n",
"u1s = sp.Symbol('u^{t-1}_{x}')\n",
"u2s = sp.Symbol('u^{t-2}_{x}')\n",
"u0sdp1 = sp.Symbol('u^{t}_{x+1}')\n",
"u0sdm1 = sp.Symbol('u^{t}_{x-1}')\n",
"u1sdp1 = sp.Symbol('u^{t-1}_{x+1}')\n",
"u1sdm1 = sp.Symbol('u^{t-1}_{x-1}')\n",
"\n",
"B0sdm1 = sp.Symbol('B^{t}_{x-1}')\n",
"B0sdp1 = sp.Symbol('B^{t}_{x+1}')\n",
"B1sdm1 = sp.Symbol('B^{t-1}_{x-1}')\n",
"B1sdp1 = sp.Symbol('B^{t-1}_{x+1}')\n",
"\n",
"C0s = sp.Symbol('C^{t}_{x}')\n",
"C0sdm1 = sp.Symbol('C^{t}_{x-1}')\n",
"C0sdp1 = sp.Symbol('C^{t}_{x+1}')\n",
"C1s = sp.Symbol('C^{t-1}_{x}')\n",
"C1sdm1 = sp.Symbol('C^{t-1}_{x-1}')\n",
"C1sdp1 = sp.Symbol('C^{t-1}_{x+1}')\n",
"\n",
"La0s = sp.Symbol('\\\\Lambda^{t}_{x}')\n",
"La0sdm1 = sp.Symbol('\\\\Lambda^{t}_{x-1}')\n",
"La0sdp1 = sp.Symbol('\\\\Lambda^{t}_{x+1}')\n",
"La1s = sp.Symbol('\\\\Lambda^{t-1}_{x}')\n",
"La1sdm1 = sp.Symbol('\\\\Lambda^{t-1}_{x-1}')\n",
"La1sdp1 = sp.Symbol('\\\\Lambda^{t-1}_{x+1}')"
]
},
{
"cell_type": "code",
"execution_count": 123,
"id": "a6a1f39a-0d04-42b0-a09b-f2f2514e77fd",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle u^{t+1}_{x} = B^{t-1}_{x+1} \\left(- \\frac{s_{1} s_{2}}{2} + \\frac{s_{1}}{2}\\right) + B^{t-1}_{x-1} \\left(\\frac{s_{1} s_{2}}{2} - \\frac{s_{1}}{2}\\right) - \\frac{B^{t}_{x+1} s_{1}}{2} + \\frac{B^{t}_{x-1} s_{1}}{2} + \\frac{s_{2} \\left(C^{t}_{x+1} + 3 \\Lambda^{t}_{x+1} - 2 u^{t}_{x+1}\\right)}{6} + \\frac{s_{2} \\left(C^{t}_{x-1} + 3 \\Lambda^{t}_{x-1} - 2 u^{t}_{x-1}\\right)}{6} - \\frac{s_{2} \\left(C^{t}_{x} + 3 \\Lambda^{t}_{x} - 2 u^{t}_{x}\\right)}{3} + u^{t-1}_{x+1} \\left(- \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} - 1\\right) + u^{t-1}_{x-1} \\left(- \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} - 1\\right) + u^{t-1}_{x} \\left(- \\frac{s_{1} s_{2}}{3} + s_{1} + \\frac{s_{2}}{3} - 1\\right) + u^{t-2}_{x} \\left(s_{1} - 1\\right) \\left(s_{2} - 1\\right) + u^{t}_{x+1} \\left(- \\frac{s_{1}}{2} - \\frac{s_{2}}{6} + 1\\right) + u^{t}_{x-1} \\left(- \\frac{s_{1}}{2} - \\frac{s_{2}}{6} + 1\\right) + u^{t}_{x} \\left(1 - \\frac{2 s_{2}}{3}\\right) + \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) \\left(C^{t-1}_{x+1} + 3 \\Lambda^{t-1}_{x+1} - 2 u^{t-1}_{x+1}\\right) + \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) \\left(C^{t-1}_{x-1} + 3 \\Lambda^{t-1}_{x-1} - 2 u^{t-1}_{x-1}\\right) + \\left(\\frac{s_{1} s_{2}}{3} - \\frac{s_{2}}{3}\\right) \\left(C^{t-1}_{x} + 3 \\Lambda^{t-1}_{x} - 2 u^{t-1}_{x}\\right)$"
],
"text/plain": [
"Eq(u^{t+1}_{x}, B^{t-1}_{x+1}*(-s_1*s_2/2 + s_1/2) + B^{t-1}_{x-1}*(s_1*s_2/2 - s_1/2) - B^{t}_{x+1}*s_1/2 + B^{t}_{x-1}*s_1/2 + s_2*(C^{t}_{x+1} + 3*\\Lambda^{t}_{x+1} - 2*u^{t}_{x+1})/6 + s_2*(C^{t}_{x-1} + 3*\\Lambda^{t}_{x-1} - 2*u^{t}_{x-1})/6 - s_2*(C^{t}_{x} + 3*\\Lambda^{t}_{x} - 2*u^{t}_{x})/3 + u^{t-1}_{x+1}*(-s_1*s_2/3 + s_1/2 + 5*s_2/6 - 1) + u^{t-1}_{x-1}*(-s_1*s_2/3 + s_1/2 + 5*s_2/6 - 1) + u^{t-1}_{x}*(-s_1*s_2/3 + s_1 + s_2/3 - 1) + u^{t-2}_{x}*(s_1 - 1)*(s_2 - 1) + u^{t}_{x+1}*(-s_1/2 - s_2/6 + 1) + u^{t}_{x-1}*(-s_1/2 - s_2/6 + 1) + u^{t}_{x}*(1 - 2*s_2/3) + (-s_1*s_2/6 + s_2/6)*(C^{t-1}_{x+1} + 3*\\Lambda^{t-1}_{x+1} - 2*u^{t-1}_{x+1}) + (-s_1*s_2/6 + s_2/6)*(C^{t-1}_{x-1} + 3*\\Lambda^{t-1}_{x-1} - 2*u^{t-1}_{x-1}) + (s_1*s_2/3 - s_2/3)*(C^{t-1}_{x} + 3*\\Lambda^{t-1}_{x} - 2*u^{t-1}_{x}))"
]
},
"execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eqsubs4 = (eqsubs3.subs({m0k0s:u0s,m0k1s:u1s,m0k2s:u2s,m0k0sdp1:u0sdp1,m0k0sdm1:u0sdm1,m0k1sdm1:u1sdm1,m0k1sdp1:u1sdp1})\n",
" .subs({meq1k0sdp1:B0sdp1,meq1k0sdm1:B0sdm1,meq1k1sdp1:B1sdp1,meq1k1sdm1:B1sdm1})\n",
" .subs({meq2k0sdp1:C0sdp1+3*La0sdp1-2*u0sdp1,meq2k0sdm1:C0sdm1+3*La0sdm1-2*u0sdm1,meq2k1sdp1:C1sdp1+3*La1sdp1-2*u1sdp1,meq2k1sdm1:C1sdm1+3*La1sdm1-2*u1sdm1})\n",
" .subs({meq2k0s:C0s+3*La0s-2*u0s,meq2k1s:C1s+3*La1s-2*u1s}))\n",
" \n",
"sp.Eq(eq1.subs({m1s:u11s,gammas[3]:1}), eqsubs4)"
]
},
{
"cell_type": "markdown",
"id": "c28f43d6-bcd7-4b91-a5a5-1daabe09504d",
"metadata": {},
"source": [
"rewriting the above equation, we have:"
]
},
{
"cell_type": "code",
"execution_count": 124,
"id": "e9edc7ff-eaa8-46ac-8b22-13bfd6eed079",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle u^{t+1}_{x} = - B^{t-1}_{x+1} \\eta_{2} + B^{t-1}_{x-1} \\eta_{2} - B^{t}_{x+1} \\eta_{1} + B^{t}_{x-1} \\eta_{1} + \\alpha_{1} u^{t}_{x} + \\alpha_{2} u^{t}_{x+1} + \\alpha_{2} u^{t}_{x-1} + \\beta_{1} u^{t-1}_{x} + \\beta_{2} u^{t-1}_{x+1} + \\beta_{2} u^{t-1}_{x-1} + \\gamma u^{t-2}_{x} + \\kappa_{1} \\left(C^{t}_{x+1} + 3 \\Lambda^{t}_{x+1} - 2 u^{t}_{x+1}\\right) + \\kappa_{1} \\left(C^{t}_{x-1} + 3 \\Lambda^{t}_{x-1} - 2 u^{t}_{x-1}\\right) - 2 \\kappa_{1} \\left(C^{t}_{x} + 3 \\Lambda^{t}_{x} - 2 u^{t}_{x}\\right) + \\kappa_{2} \\left(C^{t-1}_{x+1} + 3 \\Lambda^{t-1}_{x+1} - 2 u^{t-1}_{x+1}\\right) + \\kappa_{2} \\left(C^{t-1}_{x-1} + 3 \\Lambda^{t-1}_{x-1} - 2 u^{t-1}_{x-1}\\right) - 2 \\kappa_{2} \\left(C^{t-1}_{x} + 3 \\Lambda^{t-1}_{x} - 2 u^{t-1}_{x}\\right)$"
],
"text/plain": [
"Eq(u^{t+1}_{x}, -B^{t-1}_{x+1}*\\eta_{2} + B^{t-1}_{x-1}*\\eta_{2} - B^{t}_{x+1}*\\eta_{1} + B^{t}_{x-1}*\\eta_{1} + \\alpha_{1}*u^{t}_{x} + \\alpha_{2}*u^{t}_{x+1} + \\alpha_{2}*u^{t}_{x-1} + \\beta_{1}*u^{t-1}_{x} + \\beta_{2}*u^{t-1}_{x+1} + \\beta_{2}*u^{t-1}_{x-1} + \\gamma*u^{t-2}_{x} + \\kappa_{1}*(C^{t}_{x+1} + 3*\\Lambda^{t}_{x+1} - 2*u^{t}_{x+1}) + \\kappa_{1}*(C^{t}_{x-1} + 3*\\Lambda^{t}_{x-1} - 2*u^{t}_{x-1}) - 2*\\kappa_{1}*(C^{t}_{x} + 3*\\Lambda^{t}_{x} - 2*u^{t}_{x}) + \\kappa_{2}*(C^{t-1}_{x+1} + 3*\\Lambda^{t-1}_{x+1} - 2*u^{t-1}_{x+1}) + \\kappa_{2}*(C^{t-1}_{x-1} + 3*\\Lambda^{t-1}_{x-1} - 2*u^{t-1}_{x-1}) - 2*\\kappa_{2}*(C^{t-1}_{x} + 3*\\Lambda^{t-1}_{x} - 2*u^{t-1}_{x}))"
]
},
"execution_count": 124,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alpha1 = sp.Symbol('\\\\alpha_{1}')\n",
"alpha2 = sp.Symbol('\\\\alpha_{2}')\n",
"beta1 = sp.Symbol('\\\\beta_{1}')\n",
"beta2 = sp.Symbol('\\\\beta_{2}')\n",
"eta1 = sp.Symbol('\\\\eta_{1}')\n",
"eta2 = sp.Symbol('\\\\eta_{2}')\n",
"kappa1 = sp.Symbol('\\\\kappa_{1}')\n",
"kappa2 = sp.Symbol('\\\\kappa_{2}')\n",
"gamma = sp.Symbol('\\\\gamma')\n",
"# eqsubs5 = eqsubs4.subs({s1/2:beta1,s1*s2/2-s1/2:beta2})\n",
"eqsubs5 = (eqsubs4.subs({-s1*s2/3+s1/2+5*s2/6-1:beta2,-s1*s2/3+s1+s2/3-1:beta1,-s1/2-s2/6+1:alpha2,1-2*s2/3:alpha1,(s1-1)*(s2-1):gamma})\n",
" .subs({-s1*s2/6+s2/6:kappa2,s1*s2/3-s2/3:-2*kappa2,s2/6:kappa1,-s2/3:-2*kappa1}) \n",
" .subs({s1*s2/2-s1/2:eta2,s1/2:eta1}))\n",
"sp.Eq(eq1.subs({m1s:u11s,gammas[3]:1}), eqsubs5)"
]
},
{
"cell_type": "markdown",
"id": "db061af4-61c7-4579-a4e6-8cb31422b1f7",
"metadata": {},
"source": [
"where coefficients are given by:"
]
},
{
"cell_type": "code",
"execution_count": 125,
"id": "f4bb94ce-719b-489d-9832-afd3f205aac1",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\alpha_{1} =1 - \\frac{2 s_{2}}{3},\\qquad \\alpha_{2}=- \\frac{s_{1}}{2} - \\frac{s_{2}}{6} + 1,\\qquad \\beta_{1}=- \\frac{s_{1} s_{2}}{3} + s_{1} + \\frac{s_{2}}{3} - 1,\\qquad \\beta_{2}=- \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} - 1$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\eta_{1}=\\frac{s_{1}}{2},\\qquad \\eta_{2}=\\frac{s_{1} s_{2}}{2} - \\frac{s_{1}}{2},\\qquad \\kappa_{1}=\\frac{s_{2}}{6},\\qquad \\kappa_{1}=- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6},\\qquad \\gamma=\\left(s_{1} - 1\\right) \\left(s_{2} - 1\\right)$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"display(Math(r\"\\alpha_{1} =\" + sp.latex(1-2*s2/3) +r\",\\qquad \\alpha_{2}=\"+ sp.latex(-s1/2-s2/6+1) +r\",\\qquad \\beta_{1}=\"\n",
" + sp.latex(-s1*s2/3+s1+s2/3-1) +r\",\\qquad \\beta_{2}=\"+ sp.latex(-s1*s2/3+s1/2+5*s2/6-1) ))\n",
"display(Math(r\"\\eta_{1}=\" + sp.latex(s1/2) +r\",\\qquad \\eta_{2}=\" + sp.latex(s1*s2/2-s1/2) +r\",\\qquad \\kappa_{1}=\"+ sp.latex(s2/6) \n",
" +r\",\\qquad \\kappa_{1}=\"+ sp.latex(-s1*s2/6+s2/6) +r\",\\qquad \\gamma=\"+ sp.latex((s1-1)*(s2-1)) ))"
]
},
{
"cell_type": "markdown",
"id": "430d5c11-2d98-43a0-95e6-8a84a737736c",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": []
},
"source": [
"### Linear Convective-Diffusive Equation\n",
"\n",
"Considering linear convective-diffusive equation:\n",
"\n",
"$$\n",
"\\partial_t u + \\partial_{\\alpha}c_{\\alpha}u = \\partial_{\\alpha}\\left(\\nu \\partial_{\\alpha} u \\right) \n",
"$$\n",
"\n",
"we have $B_{\\alpha}=c_{\\alpha}u$, $c_{\\alpha}$ is a constant velocity, and $C=u$. Replacing $B_{\\alpha}$ and $C$ in FD scheme obtained"
]
},
{
"cell_type": "code",
"execution_count": 126,
"id": "b66725d7-05eb-4cca-8f18-274cad49719f",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle u^{t+1}_{x} = u^{t-1}_{x+1} \\left(\\overline{c}_{\\alpha} \\left(- \\frac{s_{1} s_{2}}{2} + \\frac{s_{1}}{2}\\right) - \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} + \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right) \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) - 1\\right) + u^{t-1}_{x-1} \\left(\\overline{c}_{\\alpha} \\left(\\frac{s_{1} s_{2}}{2} - \\frac{s_{1}}{2}\\right) - \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} + \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right) \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) - 1\\right) + u^{t-1}_{x} \\left(- \\frac{s_{1} s_{2}}{3} + s_{1} + \\frac{s_{2}}{3} + \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right) \\left(\\frac{s_{1} s_{2}}{3} - \\frac{s_{2}}{3}\\right) - 1\\right) + u^{t-2}_{x} \\left(s_{1} - 1\\right) \\left(s_{2} - 1\\right) + u^{t}_{x+1} \\left(- \\frac{\\overline{c}_{\\alpha} s_{1}}{2} - \\frac{s_{1}}{2} + \\frac{s_{2} \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right)}{6} - \\frac{s_{2}}{6} + 1\\right) + u^{t}_{x-1} \\left(\\frac{\\overline{c}_{\\alpha} s_{1}}{2} - \\frac{s_{1}}{2} + \\frac{s_{2} \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right)}{6} - \\frac{s_{2}}{6} + 1\\right) + u^{t}_{x} \\left(- \\frac{s_{2} \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right)}{3} - \\frac{2 s_{2}}{3} + 1\\right)$"
],
"text/plain": [
"Eq(u^{t+1}_{x}, u^{t-1}_{x+1}*(\\overline{c}_{\\alpha}*(-s_1*s_2/2 + s_1/2) - s_1*s_2/3 + s_1/2 + 5*s_2/6 + (3*\\overline{c}_{\\alpha}**2 - 1)*(-s_1*s_2/6 + s_2/6) - 1) + u^{t-1}_{x-1}*(\\overline{c}_{\\alpha}*(s_1*s_2/2 - s_1/2) - s_1*s_2/3 + s_1/2 + 5*s_2/6 + (3*\\overline{c}_{\\alpha}**2 - 1)*(-s_1*s_2/6 + s_2/6) - 1) + u^{t-1}_{x}*(-s_1*s_2/3 + s_1 + s_2/3 + (3*\\overline{c}_{\\alpha}**2 - 1)*(s_1*s_2/3 - s_2/3) - 1) + u^{t-2}_{x}*(s_1 - 1)*(s_2 - 1) + u^{t}_{x+1}*(-\\overline{c}_{\\alpha}*s_1/2 - s_1/2 + s_2*(3*\\overline{c}_{\\alpha}**2 - 1)/6 - s_2/6 + 1) + u^{t}_{x-1}*(\\overline{c}_{\\alpha}*s_1/2 - s_1/2 + s_2*(3*\\overline{c}_{\\alpha}**2 - 1)/6 - s_2/6 + 1) + u^{t}_{x}*(-s_2*(3*\\overline{c}_{\\alpha}**2 - 1)/3 - 2*s_2/3 + 1))"
]
},
"execution_count": 126,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c = sp.Symbol('c_{\\\\alpha}')\n",
"cb = sp.Symbol('\\overline{c}_{\\\\alpha}')\n",
"leqsubs4=sp.collect(eqsubs4.subs({B0sdp1:cb*u0sdp1,B0sdm1:cb*u0sdm1,B1sdp1:cb*u1sdp1,B1sdm1:cb*u1sdm1})\n",
" .subs({C0s:u0s,La0s:u0s*cb*cb,C0sdp1:u0sdp1,La0sdp1:u0sdp1*cb*cb,C0sdm1:u0sdm1,La0sdm1:u0sdm1*cb*cb,\n",
" C1s:u1s,La1s:u1s*cb*cb,C1sdp1:u1sdp1,La1sdp1:u1sdp1*cb*cb,C1sdm1:u1sdm1,La1sdm1:u1sdm1*cb*cb}) , [u0s,u1s,u0sdp1,u0sdm1,u1sdp1,u1sdm1])\n",
"sp.Eq(eq1.subs({m1s:u11s,gammas[3]:1}), leqsubs4)"
]
},
{
"cell_type": "markdown",
"id": "51598d36-5de9-4e36-b53f-2215d74ae843",
"metadata": {},
"source": [
"where $\\overline{c}_{\\alpha} = c_{\\alpha}\\Delta x/\\Delta t$ is the constant velocity in the lattice Boltzman scale. Rewriting the above equation with coeficients, we have:"
]
},
{
"cell_type": "code",
"execution_count": 127,
"id": "33a29a8f-6822-46f2-b42a-aba3263046ae",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle u^{t+1}_{x} = \\alpha_{1} u^{t}_{x} + \\alpha_{2} u^{t}_{x-1} + \\alpha_{3} u^{t}_{x+1} + \\beta_{1} u^{t-1}_{x} + \\beta_{2} u^{t-1}_{x-1} + \\beta_{3} u^{t-1}_{x+1} + \\gamma u^{t-2}_{x}$"
],
"text/plain": [
"Eq(u^{t+1}_{x}, \\alpha_{1}*u^{t}_{x} + \\alpha_{2}*u^{t}_{x-1} + \\alpha_{3}*u^{t}_{x+1} + \\beta_{1}*u^{t-1}_{x} + \\beta_{2}*u^{t-1}_{x-1} + \\beta_{3}*u^{t-1}_{x+1} + \\gamma*u^{t-2}_{x})"
]
},
"execution_count": 127,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"alpha3 = sp.Symbol('\\\\alpha_{3}')\n",
"beta3 = sp.Symbol('\\\\beta_{3}')\n",
"leqsubs5=(leqsubs4.subs({-cb*s1/2-s1/2+s2*(3*cb*cb-1)/6-s2/6+1:alpha3,cb*s1/2-s1/2+s2*(3*cb*cb-1)/6-s2/6+1:alpha2,1-s2*(3*cb*cb-1)/3-2*s2/3:alpha1,(s1-1)*(s2-1):gamma}) \n",
" .subs({cb*(-s1*s2/2+s1/2)-s1*s2/3+s1/2+5*s2/6-1+(3*cb*cb-1)*(-s1*s2/6+s2/6):beta3,cb*(s1*s2/2-s1/2)-s1*s2/3+s1/2+5*s2/6-1+(3*cb*cb-1)*(-s1*s2/6+s2/6):beta2,-s1*s2/3+s1+s2/3-1+(3*cb*cb-1)*(s1*s2/3-s2/3):beta1}) )\n",
"sp.Eq(eq1.subs({m1s:u11s,gammas[3]:1}), leqsubs5)"
]
},
{
"cell_type": "markdown",
"id": "69e2c05e-b3db-482a-b723-d82d66be4856",
"metadata": {},
"source": [
"where coefficients are given by:"
]
},
{
"cell_type": "code",
"execution_count": 128,
"id": "8475e442-46df-41ff-a973-7c9f72dd5497",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\alpha_{1} =- \\frac{s_{2} \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right)}{3} - \\frac{2 s_{2}}{3} + 1,\\qquad \\alpha_{2}=\\frac{\\overline{c}_{\\alpha} s_{1}}{2} - \\frac{s_{1}}{2} + \\frac{s_{2} \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right)}{6} - \\frac{s_{2}}{6} + 1,\\qquad \\alpha_{3}=- \\frac{\\overline{c}_{\\alpha} s_{1}}{2} - \\frac{s_{1}}{2} + \\frac{s_{2} \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right)}{6} - \\frac{s_{2}}{6} + 1$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\beta_{1}=- \\frac{s_{1} s_{2}}{3} + s_{1} + \\frac{s_{2}}{3} + \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right) \\left(\\frac{s_{1} s_{2}}{3} - \\frac{s_{2}}{3}\\right) - 1,\\qquad\\beta_{2}=\\overline{c}_{\\alpha} \\left(\\frac{s_{1} s_{2}}{2} - \\frac{s_{1}}{2}\\right) - \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} + \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right) \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) - 1$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\beta_{3}=\\overline{c}_{\\alpha} \\left(- \\frac{s_{1} s_{2}}{2} + \\frac{s_{1}}{2}\\right) - \\frac{s_{1} s_{2}}{3} + \\frac{s_{1}}{2} + \\frac{5 s_{2}}{6} + \\left(3 \\overline{c}_{\\alpha}^{2} - 1\\right) \\left(- \\frac{s_{1} s_{2}}{6} + \\frac{s_{2}}{6}\\right) - 1,\\qquad \\gamma=\\left(s_{1} - 1\\right) \\left(s_{2} - 1\\right)$"
],
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"display(Math(r\"\\alpha_{1} =\" + sp.latex(1-s2*(3*cb*cb-1)/3-2*s2/3) +r\",\\qquad \\alpha_{2}=\"+ sp.latex(cb*s1/2-s1/2+s2*(3*cb*cb-1)/6-s2/6+1) \n",
" +r\",\\qquad \\alpha_{3}=\"+ sp.latex(-cb*s1/2-s1/2+s2*(3*cb*cb-1)/6-s2/6+1) ))\n",
"display(Math(r\"\\beta_{1}=\"+ sp.latex(-s1*s2/3+s1+s2/3-1+(3*cb*cb-1)*(s1*s2/3-s2/3)) + r\",\\qquad\\beta_{2}=\"+ sp.latex(cb*(s1*s2/2-s1/2)-s1*s2/3+s1/2+5*s2/6-1+(3*cb*cb-1)*(-s1*s2/6+s2/6)) ))\n",
"display(Math(r\"\\beta_{3}=\"+ sp.latex(cb*(-s1*s2/2+s1/2)-s1*s2/3+s1/2+5*s2/6-1+(3*cb*cb-1)*(-s1*s2/6+s2/6)) + r\",\\qquad \\gamma=\"+ sp.latex((s1-1)*(s2-1)) ))"
]
},
{
"cell_type": "markdown",
"id": "0e85e733-a030-47f4-942b-3a445e52867a",
"metadata": {},
"source": [
"#### Fourth-order Convective-Diffusive Equation\n",
"\n",
"Expanding in Taylor series the FD form LB equation, we can truncate the equation with errors isolated in third and fourth-order of scale of $\\Delta x$: \n",
"\n",
"$$\n",
"\\begin{array}{c}\n",
"0 = \\displaystyle\\frac{\\Delta_{t}^{4} \\left(- 15 s_{1} s_{2} + 14 s_{1} + 14 s_{2} - 12\\right)}{24} \\frac{\\partial^{4}{u}}{\\partial{t^{4}}} - \\frac{\\Delta_{t}^{4} c_{\\alpha} s_{1} \\left(s_{2} - 1\\right)}{6}\\frac{\\partial^{4}{u}}{\\partial{x}\\partial{t^{3}}} + \\frac{\\Delta_{t}^{3} \\left(7 s_{1} s_{2} - 6 s_{1} - 6 s_{2} + 6\\right)}{6}\\frac{\\partial^{3}{u}}{\\partial{t^{3}}} + \\frac{\\Delta_{t}^{3} c_{\\alpha} s_{1} \\left(s_{2} - 1\\right)}{2} \\frac{\\partial^{3}{u}}{\\partial{x}\\partial{t^{2}}} \\\\ + \\displaystyle\\frac{\\Delta_{t}^{2} \\Delta_{x}^{2} \\left(\\frac{3 \\Delta_{t}^{2} c_{\\alpha}^{2} s_{1} s_{2}}{\\Delta_{x}^{2}} - \\frac{3 \\Delta_{t}^{2} c_{\\alpha}^{2} s_{2}}{\\Delta_{x}^{2}} + s_{1} s_{2} - 3 s_{1} - 4 s_{2} + 6\\right)}{12}\\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}} - \\frac{\\Delta_{t}^{2} \\Delta_{x}^{2} c_{\\alpha} s_{1} \\left(s_{2} - 1\\right)}{6}\\frac{\\partial^{4}{u}}{\\partial{x^{3}}\\partial{t}} + \\frac{\\Delta_{t}^{2} \\left(- 3 s_{1} s_{2} + 2 s_{1} + 2 s_{2}\\right)}{2} \\frac{\\partial^{2}{u}}{\\partial{t^{2}}} - \\Delta_{t}^{2} \\frac{\\partial^{2}{u}}{\\partial{x}\\partial{t}} c_{\\alpha} s_{1} \\left(s_{2} - 1\\right) \\\\ - \\displaystyle\\frac{\\Delta_{t} \\Delta_{x}^{2} \\left(\\frac{3 \\Delta_{t}^{2} c_{\\alpha}^{2} s_{1} s_{2}}{\\Delta_{x}^{2}} - \\frac{3 \\Delta_{t}^{2} c_{\\alpha}^{2} s_{2}}{\\Delta_{x}^{2}} + s_{1} s_{2} - 3 s_{1} - 4 s_{2} + 6\\right)}{6} \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}} + \\frac{\\Delta_{t} \\Delta_{x}^{2} c_{\\alpha} s_{1} s_{2}}{6} \\frac{\\partial^{3}{u}}{\\partial{x^{3}}} + \\Delta_{t} \\frac{\\partial{u}}{\\partial{t}} s_{1} s_{2} + \\Delta_{t} \\frac{\\partial{u}}{\\partial{x}} c_{\\alpha} s_{1} s_{2} + \\frac{\\Delta_{x}^{4} s_{2} \\left(\\frac{3 \\Delta_{t}^{2} c_{\\alpha}^{2} s_{1}}{\\Delta_{x}^{2}} - \\frac{6 \\Delta_{t}^{2} c_{\\alpha}^{2}}{\\Delta_{x}^{2}} + s_{1} - 2\\right)}{72} \\frac{\\partial^{4}{u}}{\\partial{x^{4}}} \\\\ + \\displaystyle\\frac{\\Delta_{x}^{2} s_{2} \\left(\\frac{3 \\Delta_{t}^{2} c_{\\alpha}^{2} s_{1}}{\\Delta_{x}^{2}} - \\frac{6 \\Delta_{t}^{2} c_{\\alpha}^{2}}{\\Delta_{x}^{2}} + s_{1} - 2\\right)}{6} \\frac{\\partial^{2}{u}}{\\partial{x^{2}}} ,\n",
"\\end{array}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "9ae04512-caee-4373-a1ff-0c9f41bf8226",
"metadata": {},
"source": [
"isolating $\\frac{\\partial u}{\\partial t}$:\n",
"\n",
"$$\n",
"\\begin{array}{c}\n",
"\\displaystyle\\frac{\\partial u}{\\partial t} + \\frac{\\partial{u}}{\\partial{x}} c_{\\alpha} = \\frac{\\partial^{2}{u}}{\\partial{x^{2}}} \\nu + \\Delta_{t}^{3} \\frac{\\partial^{4}{u}}{\\partial{t^{4}}} \\left(\\frac{5}{8} - \\frac{7}{12 s_{2}} - \\frac{7}{12 s_{1}} + \\frac{1}{2 s_{1} s_{2}}\\right) - \\frac{\\Delta_{t}^{3} c_{\\alpha}^{2}}{4}\\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}} + \\frac{\\Delta_{t}^{3} c_{\\alpha}^{2}}{4 s_{1}}\\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}} \\\\ + \\Delta_{t}^{3} \\displaystyle\\frac{\\partial^{4}{u}}{\\partial{x}\\partial{t^{3}}} c_{\\alpha} \\left(\\frac{1}{6} - \\frac{1}{6 s_{2}}\\right) + \\Delta_{t}^{2} \\frac{\\partial^{3}{u}}{\\partial{t^{3}}} \\left(- \\frac{7}{6} + \\frac{1}{s_{2}} + \\frac{1}{s_{1}} - \\frac{1}{s_{1} s_{2}}\\right) + \\frac{\\Delta_{t}^{2} c_{\\alpha}^{2}}{2} \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}} - \\frac{\\Delta_{t}^{2} c_{\\alpha}^{2}}{2 s_{1}} \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}} + \\Delta_{t}^{2} \\frac{\\partial^{3}{u}}{\\partial{x}\\partial{t^{2}}} c_{\\alpha} \\left(- \\frac{1}{2} + \\frac{1}{2 s_{2}}\\right) \\\\ + \\Delta_{t} \\Delta_{x}^{2} \\displaystyle\\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}} \\left(- \\frac{1}{12} + \\frac{1}{4 s_{2}} + \\frac{1}{3 s_{1}} - \\frac{1}{2 s_{1} s_{2}}\\right) + \\Delta_{t} \\Delta_{x}^{2} \\frac{\\partial^{4}{u}}{\\partial{x^{3}}\\partial{t}} c_{\\alpha} \\left(\\frac{1}{6} - \\frac{1}{6 s_{2}}\\right) - \\frac{\\Delta_{t} \\Delta_{x}^{2} c_{\\alpha}^{2}}{24} \\frac{\\partial^{4}{u}}{\\partial{x^{4}}} + \\frac{\\Delta_{t} \\Delta_{x}^{2} c_{\\alpha}^{2}}{12 s_{1}} \\frac{\\partial^{4}{u}}{\\partial{x^{4}}} \\\\ + \\Delta_{t} \\displaystyle\\frac{\\partial^{2}{u}}{\\partial{t^{2}}} \\left(\\frac{3}{2} - \\frac{1}{s_{2}} - \\frac{1}{s_{1}}\\right) - \\frac{\\Delta_{t} c_{\\alpha}^{2}}{2} \\frac{\\partial^{2}{u}}{\\partial{x^{2}}} + \\frac{\\Delta_{t} c_{\\alpha}^{2}}{s_{1}} \\frac{\\partial^{2}{u}}{\\partial{x^{2}}} + \\Delta_{t} \\frac{\\partial^{2}{u}}{\\partial{x}\\partial{t}} c_{\\alpha} \\left(1 - \\frac{1}{s_{2}}\\right) + \\Delta_{x}^{2} \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}} \\left(\\frac{1}{6} - \\frac{1}{2 s_{2}} - \\frac{2}{3 s_{1}} + \\frac{1}{s_{1} s_{2}}\\right) - \\frac{\\Delta_{x}^{2} c_{\\alpha}}{6}\\frac{\\partial^{3}{u}}{\\partial{x^{3}}} + \\frac{\\Delta_{x}^{4} \\left(- \\frac{1}{72} + \\frac{1}{36 s_{1}}\\right)}{\\Delta_{t}}\\frac{\\partial^{4}{u}}{\\partial{x^{4}}}\n",
"\\end{array}\n",
"$$(Full-fdm-convec-diff)\n",
"where $\\nu=c_{s}^{2}\\left(\\displaystyle\\frac{2-s1}{2s1}\\right)\\displaystyle\\frac{\\Delta x^{2}}{\\Delta t}$"
]
},
{
"cell_type": "markdown",
"id": "319d50f7-1f7a-459d-8334-15f063aaa762",
"metadata": {},
"source": [
"To correct reduce the above equation and isolate the third and fourth order of spatial derivative terms, we employ the relations:\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{2}u}{\\partial x\\,\\partial t} = \\nu \\displaystyle\\frac{\\partial^{3}u}{\\partial x^{3}} - c_{\\alpha} \\frac{\\partial^{2}u}{\\partial x^{2}} + \\mathcal{O}\\!\\left(\\Delta^{3}x + \\Delta t\\,\\Delta x\\right),\n",
"$$\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{2}u}{\\partial t^{2}} = \\nu^{2} \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{4}} - 2\\nu c_{\\alpha} \\frac{\\partial^{3}u}{\\partial x^{3}} + c_{\\alpha}^{2} \\frac{\\partial^{2}u}{\\partial x^{2}} + \\mathcal{O}(\\Delta t),\n",
"$$\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{3}u}{\\partial x\\,\\partial t^{2}} = -2\\nu c_{\\alpha} \\displaystyle \\frac{\\partial^{4}u}{\\partial x^{4}} + c_{\\alpha}^{2} \\frac{\\partial^{3}u}{\\partial x^{3}} + \\mathcal{O}(\\Delta x),\n",
"$$\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{3}u}{\\partial x^{2}\\partial t} = \\nu \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{4}} - c_{\\alpha} \\frac{\\partial^{3}u}{\\partial x^{3}} + \\mathcal{O}\\!\\left(\\frac{\\Delta^{4}x}{\\Delta t} + \\Delta^{2}x\\right),\n",
"$$\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{3}u}{\\partial t^{3}} = 3\\nu c_{\\alpha}^{2} \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{4}} - c_{\\alpha}^{3} \\frac{\\partial^{3}u}{\\partial x^{3}} + \\mathcal{O}\\!\\left(\\frac{\\Delta^{2}x}{\\Delta t}\\right),\n",
"$$\n",
"\n",
"$$ \n",
" \\displaystyle\\frac{\\partial^{4}u}{\\partial x\\,\\partial t^{3}} = - c_{\\alpha}^{3} \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{4}} + \\mathcal{O}\\!\\left(\\frac{1}{\\Delta x}\\right),\n",
"$$\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{3}\\partial t} = - c_{\\alpha} \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{4}} + \\mathcal{O}(\\Delta x),\n",
"$$\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{2}\\partial t^{2}} = c_{\\alpha}^{2} \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{4}} + \\mathcal{O}\\!\\left(\\frac{\\Delta^{2}x}{\\Delta t}\\right), \n",
"$$\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial^{4}u}{\\partial t^{4}} = c_{\\alpha}^{4} \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{4}} + \\mathcal{O}\\!\\left(\\frac{1}{\\Delta t}\\right),\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "be12012e-f57c-49af-88b3-3a391faee1f5",
"metadata": {},
"source": [
"notice that in the above relations, the derivativer higher or equat to fifth order are neglected. Replacing them in Eq. {eq}`Full-fdm-convec-diff`:\n",
"\n",
"$$\n",
"\\begin{array}{c}\n",
"\\displaystyle\\frac{\\partial u}{\\partial t} + \\frac{\\partial{u}}{\\partial{x}} c_{\\alpha} = \\frac{\\partial^{2}{u}}{\\partial{x^{2}}} \\nu + \\frac{\\partial^{3}{u}}{\\partial{x^{3}}} \\left(\\frac{\\Delta_{t}^{2} c_{\\alpha}^{3}}{6} - \\frac{\\Delta_{t}^{2} c_{\\alpha}^{3}}{2 s_{2}} - \\frac{\\Delta_{t}^{2} c_{\\alpha}^{3}}{2 s_{1}} + \\frac{\\Delta_{t}^{2} c_{\\alpha}^{3}}{s_{1} s_{2}} - 2 \\Delta_{t} \\nu c_{\\alpha} + \\frac{\\Delta_{t} \\nu c_{\\alpha}}{s_{2}} + \\frac{2 \\Delta_{t} \\nu c_{\\alpha}}{s_{1}} - \\frac{\\Delta_{x}^{2} c_{\\alpha}}{3} + \\frac{\\Delta_{x}^{2} c_{\\alpha}}{2 s_{2}} + \\frac{2 \\Delta_{x}^{2} c_{\\alpha}}{3 s_{1}} - \\frac{\\Delta_{x}^{2} c_{\\alpha}}{s_{1} s_{2}}\\right) + \\\\ \\displaystyle\\frac{\\partial^{4}{u}}{\\partial{x^{4}}} \\left(\\frac{5 \\Delta_{t}^{3} c_{\\alpha}^{4}}{24} - \\frac{5 \\Delta_{t}^{3} c_{\\alpha}^{4}}{12 s_{2}} - \\frac{\\Delta_{t}^{3} c_{\\alpha}^{4}}{3 s_{1}} + \\frac{\\Delta_{t}^{3} c_{\\alpha}^{4}}{2 s_{1} s_{2}} - 2 \\Delta_{t}^{2} \\nu c_{\\alpha}^{2} + \\frac{2 \\Delta_{t}^{2} \\nu c_{\\alpha}^{2}}{s_{2}} + \\frac{5 \\Delta_{t}^{2} \\nu c_{\\alpha}^{2}}{2 s_{1}} - \\frac{3 \\Delta_{t}^{2} \\nu c_{\\alpha}^{2}}{s_{1} s_{2}} - \\frac{7 \\Delta_{t} \\Delta_{x}^{2} c_{\\alpha}^{2}}{24} + \\frac{5 \\Delta_{t} \\Delta_{x}^{2} c_{\\alpha}^{2}}{12 s_{2}} + \\frac{5 \\Delta_{t} \\Delta_{x}^{2} c_{\\alpha}^{2}}{12 s_{1}} - \\frac{\\Delta_{t} \\Delta_{x}^{2} c_{\\alpha}^{2}}{2 s_{1} s_{2}} - \\frac{\\Delta_{x}^{2} \\nu}{12} - \\frac{\\Delta_{x}^{2} \\nu}{3 s_{2}} + \\frac{2 \\Delta_{x}^{2} \\nu}{3 s_{1} s_{2}} - \\frac{\\Delta_{x}^{2} \\nu}{3 s_{1}^{2}} - \\frac{\\Delta_{x}^{4}}{72 \\Delta_{t}} + \\frac{\\Delta_{x}^{4}}{36 \\Delta_{t} s_{1}}\\right)\n",
"\\end{array}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "e74c4fe0-94ab-4926-946c-bba1ea8ec9b7",
"metadata": {},
"source": [
"$$\n",
"\\begin{array}{c}\n",
"\\displaystyle\\frac{\\partial u}{\\partial t} + \\frac{\\partial{u}}{\\partial{x}} c_{\\alpha} = \\frac{\\partial^{2}{u}}{\\partial{x^{2}}} \\nu + \\frac{c_{\\alpha}\\Delta x^{2}TR_{3}}{6 s_{1}^{2}s_{2}}\\frac{\\partial^{3}{u}}{\\partial{x^{3}}} + \\frac{\\Delta x^{4}TR_{4}}{72\\Delta t s_{1}^{3}s_{2}} \\displaystyle\\frac{\\partial^{4}{u}}{\\partial{x^{4}}} \n",
"\\end{array}\n",
"$$\n",
"\n",
"where $TR_{3}$ and $TR_{4}$ are given by:"
]
},
{
"cell_type": "markdown",
"id": "f526193f-026e-4167-b850-78f45bfb25b6",
"metadata": {},
"source": [
"$$\n",
"TR_{3}=\\overline{c}_{\\alpha}^{2} s_{1} \\left(s_{1} s_{2} - 3 s_{1} - 3 s_{2} + 6\\right) + 2 \\left(s_{1} - 2\\right) \\left(s_{1} - s_{2}\\right)\n",
"$$\n",
"\n",
"$$\n",
"TR_{4}=\\overline{c}_{\\alpha}^{4} \\left(15 s_{1}^{3} s_{2} - 30 s_{1}^{3} - 24 s_{1}^{2} s_{2} + 36 s_{1}^{2}\\right) + \\overline{c}_{\\alpha}^{2} \\left(3 s_{1}^{3} s_{2} + 6 s_{1}^{3} - 48 s_{1}^{2} s_{2} + 48 s_{1}^{2} + 60 s_{1} s_{2} - 72 s_{1}\\right) + 4 s_{1}^{3} - 16 s_{1}^{2} + 4 s_{1} s_{2} + 16 s_{1} - 8 s_{2}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "62236f65-b86a-4ba4-8ead-8751988c9a3b",
"metadata": {},
"source": [
"#### Fourth-Order Convergence\n",
"\n",
"Making the term $TR_{3}$ and $TR_{4}$ equal zero and solving the equation, we have:\n",
"\n",
"$$\n",
"TR_{3}:\\qquad s_{2}=\\frac{s_{1} \\left(3 \\bar{c}_{\\alpha}^{2} s_{1} - 6 \\bar{c}_{\\alpha}^{2} - 2 s_{1} + 4\\right)}{\\bar{c}_{\\alpha}^{2} s_{1}^{2} - 3 \\bar{c}_{\\alpha}^{2} s_{1} - 2 s_{1} + 4},\n",
"$$\n",
"\n",
"$$\n",
"TR_{4}:\\qquad s_{2}=\\frac{2 s_{1} \\left(15 \\bar{c}^{4} s_{1}^{2} - 18 \\bar{c}^{4} s_{1} - 3 \\bar{c}^{2} s_{1}^{2} - 24 \\bar{c}^{2} s_{1} + 36 \\bar{c}^{2} - 2 s_{1}^{2} + 8 s_{1} - 8\\right)}{15 \\bar{c}^{4} s_{1}^{3} - 24 \\bar{c}^{4} s_{1}^{2} + 3 \\bar{c}^{2} s_{1}^{3} - 48 \\bar{c}^{2} s_{1}^{2} + 60 \\bar{c}^{2} s_{1} + 4 s_{1} - 8}\n",
"$$\n",
"\n",
"Plotting both solution for different values of $\\bar{c}_{\\alpha}$:"
]
},
{
"cell_type": "code",
"execution_count": 203,
"id": "2bccb8a5-4b0b-42ee-992d-822967baab0f",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"cbar=0.005: intersections at s1 = [0.9999937477340344]\n",
"cbar=0.01: intersections at s1 = [0.9999749637281587]\n",
"cbar=0.05: intersections at s1 = [0.9993519974731944]\n",
"cbar=0.1: intersections at s1 = [0.9971142700951519]\n"
]
},
{
"data": {
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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import numpy as np\n",
"import sympy as sp\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib\n",
"matplotlib.rcParams['mathtext.fontset'] = 'cm'\n",
"# ---------------- Symbols ----------------\n",
"s1, s2, cbar = sp.symbols('s1 s2 cbar')\n",
"A = cbar**2\n",
"\n",
"# ---------------- Eq 1: solve for s2 ----------------\n",
"s2_expr_1 = sp.solve(sp.Eq(A*s1*(s1*s2 - 3*s1 - 3*s2 + 6) + 2*(s1-2)*(s1-s2), 0), s2)[0]\n",
"\n",
"# ---------------- Eq 2: s2(s1) (your big equation) ----------------\n",
"s2_expr_2 = (2*s1*(15*cbar**4*s1**2 - 18*cbar**4*s1\n",
" - 3*cbar**2*s1**2 - 24*cbar**2*s1\n",
" + 36*cbar**2 - 2*s1**2 + 8*s1 - 8)\n",
") / (15*cbar**4*s1**3 - 24*cbar**4*s1**2\n",
" + 3*cbar**2*s1**3 - 48*cbar**2*s1**2\n",
" + 60*cbar**2*s1 + 4*s1 - 8)\n",
"\n",
"# difference g(s1) = f1 - f2 (roots => intersections)\n",
"g_expr = sp.simplify(s2_expr_1 - s2_expr_2)\n",
"\n",
"# lambdify for fast evaluation\n",
"f1 = sp.lambdify((s1, cbar), s2_expr_1, \"numpy\")\n",
"f2 = sp.lambdify((s1, cbar), s2_expr_2, \"numpy\")\n",
"g = sp.lambdify((s1, cbar), g_expr, \"numpy\")\n",
"\n",
"# numeric root refinement (mpmath via sympy)\n",
"def refine_root(cb, x0):\n",
" x = sp.nsolve(g_expr.subs(cbar, cb), s1, x0)\n",
" return float(x)\n",
"\n",
"def unique_sorted(xs, tol=1e-6):\n",
" xs = sorted(xs)\n",
" out = []\n",
" for x in xs:\n",
" if not out or abs(x - out[-1]) > tol:\n",
" out.append(x)\n",
" return out\n",
"\n",
"# ---------------- Parameters ----------------\n",
"cbar_vals = [0.005, 0.01, 0.05, 0.1]\n",
"s1_min, s1_max = 0.01, 3.0\n",
"Nscan = 4000\n",
"s1_grid = np.linspace(s1_min, s1_max, Nscan)\n",
"\n",
"# ---------------- Plot ----------------\n",
"plt.figure(figsize=(9, 5))\n",
"\n",
"for cb in cbar_vals:\n",
" y1 = f1(s1_grid, cb)\n",
" y2 = f2(s1_grid, cb)\n",
" yg = g(s1_grid, cb)\n",
"\n",
" # mask blow-ups / poles\n",
" y1 = np.where(np.isfinite(y1) & (np.abs(y1) < 10), y1, np.nan)\n",
" y2 = np.where(np.isfinite(y2) & (np.abs(y2) < 10), y2, np.nan)\n",
" yg = np.where(np.isfinite(yg) & (np.abs(yg) < 1e6), yg, np.nan)\n",
"\n",
" # draw both curves\n",
" plt.plot(s1_grid, y1, linestyle='-', label=rf\"$TR_{3}$, $\\bar c_\\alpha={cb}$\")\n",
" plt.plot(s1_grid, y2, linestyle='--', label=rf\"$TR_{4}$, $\\bar c_\\alpha={cb}$\")\n",
"\n",
" # --- find sign changes of g on the scan grid ---\n",
" roots_guess = []\n",
" for i in range(len(s1_grid) - 1):\n",
" a, b = yg[i], yg[i+1]\n",
" if np.isnan(a) or np.isnan(b):\n",
" continue\n",
" if a == 0.0:\n",
" roots_guess.append(s1_grid[i])\n",
" elif a * b < 0: # sign change\n",
" roots_guess.append(0.5 * (s1_grid[i] + s1_grid[i+1]))\n",
"\n",
" # refine guesses with nsolve\n",
" roots = []\n",
" for x0 in roots_guess:\n",
" try:\n",
" xr = refine_root(cb, x0)\n",
" if s1_min <= xr <= s1_max:\n",
" roots.append(xr)\n",
" except Exception:\n",
" pass\n",
"\n",
" roots = unique_sorted(roots, tol=1e-5)\n",
"\n",
" # compute s2 at intersection points and plot markers\n",
" for xr in roots:\n",
" yr = float(sp.N(s2_expr_1.subs({cbar: cb, s1: xr})))\n",
" if np.isfinite(yr) and abs(yr) < 10:\n",
" plt.plot([xr], [yr], marker='o', linestyle='None')\n",
"\n",
" if roots:\n",
" print(f\"cbar={cb}: intersections at s1 =\", roots)\n",
"\n",
"plt.xlabel(r\"$s_1$\",fontsize=16)\n",
"plt.ylabel(r\"$s_2$\",fontsize=16)\n",
"plt.xlim(0, 2)\n",
"plt.ylim(0, 3)\n",
"plt.grid(True)\n",
"plt.legend(ncol=2, fontsize=9)\n",
"plt.tight_layout()\n",
"plt.show()\n"
]
},
{
"cell_type": "code",
"execution_count": 187,
"id": "4dfb3e2f-118a-47d0-98ee-f0b64e41cc0b",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input",
"hide-output"
]
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dx=6.2832e-01,\t dt=3.1416e-01\t nt=77,\t tn=24.19\n",
"c_lbm=0.006,\t tau1=1.0000,\t tau2=1.0000\n",
"s1=1.0000,\t s2=1.0000\n",
"nt=77.000\t, nue=0.1667\n",
"dx=3.1416e-01,\t dt=7.8540e-02\t nt=306,\t tn=24.03\n",
"c_lbm=0.003,\t tau1=1.0000,\t tau2=1.0000\n",
"s1=1.0000,\t s2=1.0000\n",
"nt=306.000\t, nue=0.1667\n",
"dx=1.5708e-01,\t dt=1.9635e-02\t nt=1223,\t tn=24.01\n",
"c_lbm=0.002,\t tau1=1.0000,\t tau2=1.0000\n",
"s1=1.0000,\t s2=1.0000\n",
"nt=1223.000\t, nue=0.1667\n",
"dx=7.8540e-02,\t dt=4.9087e-03\t nt=4890,\t tn=24.00\n",
"c_lbm=0.001,\t tau1=1.0000,\t tau2=1.0000\n",
"s1=1.0000,\t s2=1.0000\n",
"nt=4890.000\t, nue=0.1667\n"
]
}
],
"source": [
"import warnings\n",
"warnings.filterwarnings(\"ignore\")\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib\n",
"matplotlib.rcParams['mathtext.fontset'] = 'cm'\n",
"matplotlib.rcParams['mathtext.fontset'] = 'cm'\n",
"#--------------------------------------- Parameters (physical units) ------------------------------------------\n",
"A = 1.0 # Amplitude\n",
"phi = 0.0 # Wave initial shift\n",
"k_w = 1.0 # Wave number\n",
"c = 0.0125 # wave speed\n",
"nu = 0.066666666666666666*np.pi # Diffusive coefficient\n",
"m_waves = 1 # Wave Period\n",
"L = m_waves * (2.0*np.pi / k_w) # Domain Lentgh\n",
"t_end = 24.0 # Final time\n",
"#--------------------------------------- Lattice-Properties-D1Q3 ----------------------------------------------\n",
"w = np.array([4.0/6.0, 1.0/6.0, 1.0/6.0],dtype=\"float64\")\n",
"cx = np.array([0, 1, -1],dtype=\"int16\") \n",
"cs=1.0/np.sqrt(3.0);\n",
"#--------------------------------- Initialization - Save Data Arrays -----------------------------------------------\n",
"cases=4\n",
"Nx0 = np.array([10, 20, 40, 80],dtype=\"int64\")\n",
"r0 = 2**(0)*np.array([2.0**(1), 2.0**(2), 2.0**(3), 2.0**(4)],dtype=\"float64\")\n",
"u_cases = np.empty(cases, dtype=object) # array field used to same data over time\n",
"for i in range(cases):\n",
" u_cases[i] = np.zeros((Nx0[i]), dtype=\"float64\")\n",
"for case in range(0,cases):\n",
" #------------------------------------- Parameters (numerical units) -------------------------------------------\n",
" Nx=Nx0[case] # Numerical Length\n",
" x = np.linspace(0.0, L, Nx, endpoint=False,dtype=\"float64\") # Numerical Domain\n",
" dx = L / Nx # Grid size\n",
" r = 1*r0[case] # dx/dt relation\n",
" dt = dx/r\n",
" ce = c/r\n",
" nue = nu * dt / (dx*dx)\n",
" tau1 = 0.5 + nue / cs**2\n",
" om1=1.0/tau1\n",
" om2=om1*(3.0*ce*ce*om1 - 6.0*ce*ce - 2.0*om1 +4.0)/(ce*ce*om1*om1 - 3.0*ce*ce*om1 - 2.0*om1 +4.0)\n",
" tau2=1.0/om2\n",
" nt = int(np.ceil(t_end / dt))\n",
" print(f\"dx={dx:.4e},\\t dt={dt:.4e}\\t nt={nt:d},\\t tn={nt*dt:.2f}\") # Print values for check\n",
" print(f\"c_lbm={ce:.3f},\\t tau1={tau1:.4f},\\t tau2={tau2:.4f}\")\n",
" print(f\"s1={1.0/tau1:.4f},\\t s2={1.0/tau2:.4f}\")\n",
" print(f\"nt={nt:.3f}\\t, nue={nue:.4f}\")\n",
" #--------------------------------- Initialization - LBM - Numerical Arrays -----------------------------------------\n",
" u=np.zeros((Nx),dtype=\"float64\")\n",
" u = A * np.sin(k_w*(x+dx/2) + phi)\n",
" mx=np.zeros((Nx),dtype=\"float64\")\n",
" mxx=np.zeros((Nx),dtype=\"float64\")\n",
" f=np.zeros((3,Nx),dtype=\"float64\")\n",
" feq=np.zeros((3,Nx),dtype=\"float64\")\n",
" fp=np.zeros((3,Nx),dtype=\"float64\")\n",
" for k in range(0,3):\n",
" f[k,:]=w[k]*u[:]*(1.0 + cx[k]*ce/cs**2 + ce*ce*(cx[k]**2-cs**2)/(2.0*cs**4))\n",
" fp[k,:]=w[k]*u[:]*(1.0 + cx[k]*ce/cs**2 + ce*ce*(cx[k]**2-cs**2)/(2.0*cs**4))\n",
" #----------------------------------------- Init Loop --------------------------------------------------------\n",
" for t in range(nt*10):\n",
" #========================================= LBM - Solution =============================================\n",
" #--------------------Collision----------------\n",
" for k in range(0,3):\n",
" feq[k,:]= w[k]*u[:]*(1.0 + cx[k]*ce/cs**2 + ce*ce*(cx[k]**2-cs**2)/(2.0*cs**4))\n",
" mx = np.einsum('i,ix->x', cx, f-feq)\n",
" mxx = np.einsum('i,ix->x', 3.0*cx*cx-2.0, f-feq)\n",
" for k in range(0,3):\n",
" fp[k,:]= feq[k,:] + w[k]*( (1.0-1.0/tau1)*cx[k]*mx[:]/cs**2 + (1.0-1.0/tau2)*(cx[k]**2-cs**2)*mxx[:]/(2.0*cs**2) )\n",
" #-----------------streaming-------------------\n",
" for k in range(0,3):\n",
" f[k,:]=np.roll(fp[k,:], cx[k], axis=0)\n",
" #----------------------------------------- Maind Loop --------------------------------------------------------\n",
" for t in range(nt):\n",
" #========================================= LBM - Solution =============================================\n",
" #--------------------Collision----------------\n",
" for k in range(0,3):\n",
" feq[k,:]= w[k]*(u[:]+cx[k]*(ce*u[:])/cs**2 + u[:]*ce*ce*(cx[k]**2-cs**2)/(2.0*cs**4))\n",
" mx = np.einsum('i,ix->x', cx, f-feq)\n",
" mxx = np.einsum('i,ix->x', 3.0*cx*cx-2.0, f-feq)\n",
" for k in range(0,3):\n",
" # fp[k,:]= f[k,:] - w[k]*( cx[k]*mx[:]/(tau1*cs**2) + (cx[k]**2-cs**2)*mxx[:]/(2.0*cs**2*tau2) )\n",
" fp[k,:]= feq[k,:] + w[k]*( (1.0-1.0/tau1)*cx[k]*mx[:]/cs**2 + (1.0-1.0/tau2)*(cx[k]**2-cs**2)*mxx[:]/(2.0*cs**2) )\n",
" #-----------------streaming-------------------\n",
" for k in range(0,3):\n",
" f[k,:]=np.roll(fp[k,:], cx[k], axis=0)\n",
" #----------------------Macro------------------\n",
" # u[:]=f[0,:]+f[1,:]+f[2,:]\n",
" u=np.einsum('ix->x', f)\n",
" #---------------------save-field-cases--------------\n",
" u_cases[case][:]=u[:]"
]
},
{
"cell_type": "code",
"execution_count": 188,
"id": "85b801cc-ee77-4f66-9e11-5137b695900a",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"#****************************************Data-Analysis*********************************************\n",
"# ---------------- exact solution ----------------\n",
"def u_exact(x, t, A, k_w, phi, c, nu):\n",
" # print(f\"dx={dx:.4e},\\t dt={dt:.4e}\\t nt={nt:d},\\t tn={nt*dt:.2f}\")\n",
" return A * np.exp(-nu * k_w**2 * t) * np.sin(k_w * ((x+dx/2) - c*t) + phi)\n",
"u_ana = np.empty(cases, dtype=object) # array field used to same data over time\n",
"xl = np.empty(cases, dtype=object) # array field used to same data over time\n",
"for i in range(cases):\n",
" Nx=Nx0[i] # Numerical Length\n",
" x = np.linspace(0.0, L, Nx, endpoint=False,dtype=\"float64\") # Numerical Domain\n",
" dx = L / Nx # Grid size\n",
" r = r0[i] # dx/dt relation\n",
" dt = dx/r\n",
" nt = int(np.ceil(t_end / dt))\n",
" xl[i] = np.linspace(0.0, L, Nx0[i], endpoint=False,dtype=\"float64\")\n",
" u_ana[i] = np.zeros((Nx0[i]), dtype=\"float64\")\n",
" u_ana[i] = u_exact(xl[i], nt*dt , A, k_w, phi, c, nu)\n",
"# -----------------Plot Results --------------------\n",
"plt.figure(figsize=(10,5))\n",
"colors = plt.cm.viridis(np.linspace(0, 1, cases))\n",
"for i in range(cases):\n",
" plt.plot(xl[i], u_cases[i], \"s\", color=colors[i], lw=1, label=f\"LBM $Nx={Nx0[i]:.0f}$\",fillstyle='none')\n",
" plt.plot(xl[i], u_ana[i], color=colors[i], lw=1, label=f\"Exact $t={t_end:.2f}$\")\n",
"\n",
"plt.xlabel(\"$x$\",fontsize=16)\n",
"plt.ylabel(\"$u(x,t)$\",fontsize=16)\n",
"plt.title(\"1D Convection-Diffusion: LBM vs Analytical Solution\")\n",
"plt.grid(True, alpha=0.3)\n",
"plt.legend(ncol=5,fontsize=9,loc=\"center left\",bbox_to_anchor=(0.0, 1.2))\n",
"plt.tight_layout()\n",
"plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 190,
"id": "1ac3ec21-5d5f-4feb-a9c5-7308a968ef97",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input",
"hide-output"
]
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Erro0= 0.0015336318570353604\n",
"Erro1= 9.266539911212256e-05\n",
"Erro2= 5.748285219599088e-06\n",
"Erro3= 3.5852375607247746e-07\n",
"[-4.01986148 2.76697863]\n",
"ltests[i] = np.array([1.00,0.0125,4.02,0.0015336318570353604,9.266539911212256e-05,5.748285219599088e-06,3.5852375607247746e-07],dtype=\"float64\")\n"
]
}
],
"source": [
"Erro= np.zeros((cases), dtype=\"float64\")\n",
"for i in range(cases):\n",
" Erro[i]=np.sqrt(np.sum((u_cases[i]-u_ana[i])**2))/np.sqrt(np.sum((u_ana[i])**2))\n",
" print(f'Erro{i}=',Erro[i])\n",
"TEp=np.polyfit(np.log(Nx0), np.log(Erro), 1)\n",
"print(TEp)\n",
"print(f'ltests[i] = np.array([{1/tau1:.2f},{c:.4f},{-TEp[0]:.2f},{Erro[0]},{Erro[1]},{Erro[2]},{Erro[3]}],dtype=\"float64\")')"
]
},
{
"cell_type": "code",
"execution_count": 191,
"id": "71c5ec88-be06-4d19-8011-db5e6f321739",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [],
"source": [
"#------------------ List of teste cases ---------------------------\n",
"ntests=4\n",
"ltests = np.empty(ntests, dtype=object) # list [omega, cs**2, a, erro1, erro2, erro3]\n",
"ltests[0] = np.array([1.00,1.00,4.57,0.15612287609561742,0.004397128432778937,0.00019967538064166106,1.135259052408131e-05],dtype=\"float64\")\n",
"ltests[1] = np.array([1.00,0.50,4.20,0.02098923999495575,0.0009645322589991023,5.511609751733689e-05,3.3643885708974955e-06],dtype=\"float64\")\n",
"ltests[2] = np.array([1.00,0.25,4.06,0.006086286711464546,0.0003438989164574731,2.0955010980109768e-05,1.3010601273425052e-06],dtype=\"float64\")\n",
"ltests[3] = np.array([1.00,0.0125,4.02,0.0015336318570353604,9.266539911212256e-05,5.748285219599088e-06,3.5852375607247746e-07],dtype=\"float64\")"
]
},
{
"cell_type": "code",
"execution_count": 200,
"id": "57034fd1-c82c-44d0-acdc-805949ceffa7",
"metadata": {
"editable": true,
"slideshow": {
"slide_type": ""
},
"tags": [
"hide-input"
]
},
"outputs": [
{
"data": {
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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"matplotlib.rcParams['mathtext.fontset'] = 'cm'\n",
"plt.figure(figsize=(7,4))\n",
"plt.title(\"$Grid$ $Convergence$ $Test$\",fontsize=14)\n",
"colors = plt.cm.viridis(np.linspace(0, 1, ntests))\n",
"# plt.loglog(1/Nx0,ltests[0][3:],'r--',fillstyle='none', label=f'$\\\\omega={ltests[0][0]}$ - $Slope={ltests[0][2]}$')\n",
"for i in range(4):\n",
" plt.loglog(1/Nx0,ltests[i][3:],'--',color=colors[i],fillstyle='none', label=f'$\\\\omega_{1}={ltests[i][0]}$ - $Slope={ltests[i][2]}$')\n",
" \n",
"plt.loglog(1/Nx0,1.*2000.0/(Nx0**4),'k-',fillstyle='none')\n",
"plt.text(0.02, 0.0007, \"$4th-order$ $Slope$\", rotation=18,fontsize=12)\n",
"# plt.ylim(0,0.01)\n",
"plt.grid(True, alpha=0.3)\n",
"# plt.legend(loc=2,ncol=3,fontsize=8.5)\n",
"plt.legend(ncol=3,fontsize=9,loc=\"center left\",bbox_to_anchor=(0.0, 1.2))\n",
"plt.ylabel('$E_{T}$',fontsize=16,rotation=0,horizontalalignment='right')\n",
"plt.xlabel('$\\Delta x$',fontsize=16)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "210f8ac0-cb76-4141-b15c-94a53dc00f5a",
"metadata": {},
"source": [
"### Non-Linear Convective Equation"
]
},
{
"cell_type": "markdown",
"id": "c9186ce0-f664-4ffd-9707-5db49b31d618",
"metadata": {},
"source": [
"Expanding in Taylor series the FD form LB equation, we can truncate the equation in second-order errors of $\\Delta x$-scale: \n",
"\n",
"$$\n",
"0=- \\Delta_{t}^{2} \\frac{\\partial^{2}{B}}{\\partial{x}\\partial{t}} s_{1} \\left(s_{2} - 1\\right) + \\frac{\\Delta_{t}^{2} \\frac{\\partial^{2}{u}}{\\partial{t^{2}}} \\left(- 3 s_{1} s_{2} + 2 s_{1} + 2 s_{2}\\right)}{2} + \\Delta_{t} \\frac{\\partial{B}}{\\partial{x}} s_{1} s_{2} + \\Delta_{t} \\frac{\\partial{u}}{\\partial{t}} s_{1} s_{2} + \\frac{\\Delta_{x}^{2} s_{2} \\left(\\frac{3 \\Delta_{t}^{2} \\frac{\\partial^{2}{\\Lambda}}{\\partial{x^{2}}} s_{1}}{\\Delta_{x}^{2}} - \\frac{6 \\Delta_{t}^{2} \\frac{\\partial^{2}{\\Lambda}}{\\partial{x^{2}}}}{\\Delta_{x}^{2}} \\right)}{6}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "3d5b6dd6-adaa-4420-b6c5-1693140ad7f7",
"metadata": {},
"source": [
"isolating $\\frac{\\partial u}{\\partial t}$:\n",
"\n",
"$$\n",
"\\frac{\\partial u}{\\partial t}=\\Delta_{t} \\frac{\\partial^{2}{B}}{\\partial{x}\\partial{t}} \\left(1 - \\frac{1}{s_{2}}\\right) + \\Delta_{t} \\frac{\\partial^{2}{\\Lambda}}{\\partial{x^{2}}} \\left(- \\frac{1}{2} + \\frac{1}{s_{1}}\\right) + \\Delta_{t} \\frac{\\partial^{2}{u}}{\\partial{t^{2}}} \\left(\\frac{3}{2} - \\frac{1}{s_{2}} - \\frac{1}{s_{1}}\\right) - \\frac{\\partial{B}}{\\partial{x}} \n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "7aef87b9-fe6b-4628-a581-ba6193c25f52",
"metadata": {},
"source": [
"To correct reduce the above equation and isolate the second-order of spatial derivative terms, we employ the relations:\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial u}{\\partial t} = - \\frac{\\partial B}{\\partial x} \n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "d4b194f1-e51f-4eb1-b341-12e6c5411938",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{2}u}{\\partial t^{2}} = - \\frac{\\partial^{2}B}{\\partial x\\partial t} = -\\frac{\\partial }{\\partial x}\\left( \\frac{\\partial u}{\\partial t}B_{u} \\right) = - \\frac{\\partial }{\\partial x}\\left[ B_{u} \\frac{\\partial B}{\\partial x} \\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "77a3d6ba-8331-4cbd-be5c-d0124eacfdfc",
"metadata": {},
"source": [
"replacing the above relations in the second-order expanded equation:\n",
"\n",
"$$\n",
"\\frac{\\partial u}{\\partial t}=\\Delta_{t} \\frac{\\partial^{2}{B}}{\\partial{x}\\partial{t}} \\left(- \\frac{1}{2} + \\frac{1}{s_{1}}\\right) + \\Delta_{t} \\frac{\\partial^{2}{\\Lambda}}{\\partial{x^{2}}} \\left(- \\frac{1}{2} + \\frac{1}{s_{1}}\\right) - \\frac{\\partial{B}}{\\partial{x}}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "8a74dca4-8e8f-40aa-8186-e5f17ca7d39d",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dx= 0.8 \t dt= 0.2\n"
]
},
{
"data": {
"image/png": 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",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from pylab import *\n",
"matplotlib.rcParams['mathtext.fontset'] = 'cm'\n",
"#***************************************InputParameters************************************************\n",
"Nx=26 #Square Domain Length\n",
"#***************************************Lattice-Properties-D1Q3*************************************************\n",
"cs=1.0/sqrt(3.0);\n",
"w = np.array([4.0/6.0, 1.0/6.0, 1.0/6.0],dtype=\"float64\")\n",
"cx = np.array([0, 1, -1],dtype=\"int16\") \n",
"#***************************************LBM-Scale************************************************\n",
"L = 20.0 # Length of the reservoir\n",
"T = 10.0 # Total simulation time\n",
"dx = L / (Nx-1) # Spatial step size\n",
"c=2**(2) # c=dx/dt\n",
"dt=dx/c # Time step size\n",
"nt = int(T / dt) # Time step number\n",
"uo=1.0 # Constant Fluid Flux\n",
"m=1.0 # Dynamic Viscosity ration\n",
"print('dx=',dx,'\\t dt=',dt)\n",
"#***************************************Initialization************************************************\n",
"ue=uo/c\n",
"sk=np.zeros((Nx),dtype=\"float64\")\n",
"Bx=np.zeros((Nx),dtype=\"float64\")\n",
"f=np.zeros((3,Nx),dtype=\"float64\")\n",
"for k in range(0,3):\n",
" f[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]*3.0)\n",
"#***************************************Loop************************************************\n",
"for t in range(nt):\n",
" #----------------------Macro------------------\n",
" sk[:]=f[0,:]+f[1,:]+f[2,:]\n",
" Bx=ue*sk*sk/(sk*sk+(1.0-sk)**(2.0))\n",
" #--------------------Collision----------------\n",
" for k in range(0,3):\n",
" f[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]/cs**2)\n",
" #-----------------streaming-------------------\n",
" for k in range(0,3):\n",
" f[k,:]=np.roll(f[k,:], cx[k], axis=0)\n",
" #-----------------Boundaries-----------------------\n",
" f[1,0]= 1.0 - f[0,0]-f[2,0]\n",
" f[0,Nx-1]=f[0,Nx-2]\n",
" f[1,Nx-1]=f[1,Nx-2]\n",
" f[2,Nx-1]=f[2,Nx-2]\n",
"x = np.linspace(0, L, Nx)\n",
"plt.plot(x, sk, 'k-' ,label='Water Saturation $(t=10)$')\n",
"plt.xlabel(r\"$x$\",fontsize=18)\n",
"plt.ylabel(r\"$u(x,t)$\",fontsize=18)\n",
"plt.xlim(0,20)\n",
"plt.ylim(0,1)\n",
"plt.legend()\n",
"plt.grid(True)\n",
"plt.tight_layout()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "c67b146f-bb18-4623-aa07-de348a8495ac",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"%pylab is deprecated, use %matplotlib inline and import the required libraries.\n",
"Populating the interactive namespace from numpy and matplotlib\n"
]
},
{
"name": "stderr",
"output_type": "stream",
"text": [
"/usr/local/lib/python3.10/dist-packages/IPython/core/magics/pylab.py:166: UserWarning: pylab import has clobbered these variables: ['f']\n",
"`%matplotlib` prevents importing * from pylab and numpy\n",
" warn(\"pylab import has clobbered these variables: %s\" % clobbered +\n"
]
},
{
"ename": "NameError",
"evalue": "name 'tau' is not defined",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn[4], line 57\u001b[0m\n\u001b[1;32m 54\u001b[0m fneq[k,:]\u001b[38;5;241m=\u001b[39mw[k]\u001b[38;5;241m*\u001b[39m(cx[k]\u001b[38;5;241m*\u001b[39mPix[:]\u001b[38;5;241m*\u001b[39m\u001b[38;5;241m3.0\u001b[39m)\n\u001b[1;32m 55\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m k \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(\u001b[38;5;241m0\u001b[39m,\u001b[38;5;241m3\u001b[39m):\n\u001b[1;32m 56\u001b[0m \u001b[38;5;66;03m#f[k,:]=feq[k,:] +(1.0-1.0/tau)*fneq[k,:]\u001b[39;00m\n\u001b[0;32m---> 57\u001b[0m f[k,:]\u001b[38;5;241m=\u001b[39mf[k,:] \u001b[38;5;241m-\u001b[39m(\u001b[38;5;241m1.0\u001b[39m\u001b[38;5;241m/\u001b[39m\u001b[43mtau\u001b[49m)\u001b[38;5;241m*\u001b[39m(f[k,:]\u001b[38;5;241m-\u001b[39mfeq[k,:])\n\u001b[1;32m 58\u001b[0m \u001b[38;5;66;03m#f[k,:]=feq[k,:] \u001b[39;00m\n\u001b[1;32m 59\u001b[0m \u001b[38;5;66;03m#-----------------streaming-------------------\u001b[39;00m\n\u001b[1;32m 60\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m k \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mrange\u001b[39m(\u001b[38;5;241m0\u001b[39m,\u001b[38;5;241m3\u001b[39m):\n",
"\u001b[0;31mNameError\u001b[0m: name 'tau' is not defined"
]
}
],
"source": [
"%pylab inline\n",
"#***************************************Allocating Arrys************************************************\n",
"#--------------------------------------------Int--------------------------------------------------------\n",
"n=int(Nx);\n",
"cx=np.ones(3,dtype=\"int8\")*0;\n",
"#------------------------------------------Double-------------------------------------------------------\n",
"w=np.ones(3,dtype=\"float64\")*0\n",
"f=np.ones((3,n),dtype=\"float64\")*0;\n",
"feq=np.ones((3,n),dtype=\"float64\")*0;\n",
"fneq=np.ones((3,n),dtype=\"float64\")*0;\n",
"sk=np.ones((n),dtype=\"float64\")*0\n",
"Bx=np.ones((n),dtype=\"float64\")*0\n",
"Pix=np.ones((n),dtype=\"float64\")*0\n",
"La=np.ones((n),dtype=\"float64\")*0\n",
"dsdx=np.ones((n),dtype=\"float64\")*0\n",
"#********************************************Init-Parameters*********************************************\n",
"cs=1.0/sqrt(3.0);\n",
"w[0] = 4.0 / 6.0; w[1] = 1.0 / 6.0; w[2] = 1.0 / 6.0;\n",
"cx[0] = 0; cx[1] = 1; cx[2] = -1;\n",
"uo=ue\n",
"#---------------------Initialization-----------------------\n",
"for i in range(0,int(Nx/3)):\n",
" sk[i]=1.0-3.0*i/Nx\n",
" #sk[i]=0.0\n",
"Bx=uo*sk**(2.0)/(sk**(2.0)+(1.0-sk)**(2.0))\n",
"aa=sk*sk+(1.0-sk)**2\n",
"Gamma=uo*uo*( (sk-0.5)*(0.5-1.0/(6.0*aa)*(1.0-2.0/aa))/aa + 0.5*np.arctan(2.0*sk-1.0) )\n",
"Psi=-4.0*(sk**(3.0)/3.0-sk**(2.0)/2.0)\n",
"La=Gamma+cs*cs*Psi\n",
"for k in range(0,3):\n",
" #f[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]*3.0)\n",
" f[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]*3.0+4.5*(La[:]-sk[:]/3.0)*(cx[k]*cx[k]-1.0/3.0) )\n",
" #f[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]*3.0+4.5*(La[:])*(cx[k]*cx[k]-1.0/3.0) )\n",
"#--------------------------------MainLoop------------------------------\n",
"import time\n",
"mstep=160\n",
"start=time.time()\n",
"for kk in range(0,mstep):\n",
"#for kk in range(0,1):\n",
" #----------------------Macro------------------\n",
" sk[:]=f[0,:]+f[1,:]+f[2,:]\n",
" Bx=uo*sk*sk/(sk*sk+(1.0-sk)**(2.0))\n",
" aa=sk*sk+(1.0-sk)**2\n",
" Gamma=uo*uo*((sk-0.5)*(0.5-1.0/(6.0*aa)*(1.0-2.0/aa))/aa + 0.5*np.arctan(2.0*sk-1.0))\n",
" Psi=-4.0*(sk**(3.0)/3.0-sk**(2.0)/2.0)\n",
" La=Gamma+cs*cs*Psi\n",
" #--------------------Collision----------------\n",
" for k in range(0,3):\n",
" feq[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]*3.0+4.5*(La[:]-sk[:]/3.0)*(cx[k]*cx[k]-1.0/3.0) )\n",
" #feq[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]*3.0+4.5*(La[:])*(cx[k]*cx[k]-1.0/3.0) )\n",
" #feq[k,:]=w[k]*(sk[:]+cx[k]*Bx[:]*3.0 )\n",
" Pix= (f[1,:]-feq[1,:])*cx[1]+(f[2,:]-feq[2,:])*cx[2]\n",
" for k in range(0,3):\n",
" fneq[k,:]=w[k]*(cx[k]*Pix[:]*3.0)\n",
" for k in range(0,3):\n",
" #f[k,:]=feq[k,:] +(1.0-1.0/tau)*fneq[k,:]\n",
" f[k,:]=f[k,:] -(1.0/tau)*(f[k,:]-feq[k,:])\n",
" #f[k,:]=feq[k,:] \n",
" #-----------------streaming-------------------\n",
" for k in range(0,3):\n",
" f[k,:]=np.roll(f[k,:], cx[k], axis=0)\n",
" #-----------------Boundaries-----------------------\n",
" f[1,0]= 1.0 - f[0,0]-f[2,0]\n",
" f[0,n-1]=f[0,n-2]\n",
" f[1,n-1]=f[1,n-2]\n",
" f[2,n-1]=f[2,n-2]"
]
},
{
"cell_type": "markdown",
"id": "34c49413-440a-4d3d-b721-8af42ded78b3",
"metadata": {},
"source": [
"### Non-Linear Convective-Diffusive Equation"
]
},
{
"cell_type": "markdown",
"id": "0587402a-ab61-4a23-bf73-9edcdbbefc67",
"metadata": {},
"source": [
"Expanding in Taylor series the FD form LB equation, we can truncate the equation with errors isolated in third and fourth-order of scale of $\\Delta x$: \n",
"\n",
"$$\n",
"\\begin{array}{c}\n",
"0 = \\frac{\\Delta_{t}^{4} \\frac{\\partial^{4}{u}}{\\partial{t^{4}}} \\left(- 15 s_{1} s_{2} + 14 s_{1} + 14 s_{2} - 12\\right)}{24} - \\frac{\\Delta_{t}^{3} \\Delta_{x} \\frac{\\partial^{4}{B}}{\\partial{x}\\partial{t^{3}}} s_{1} \\left(s_{2} - 1\\right)}{6} + \\frac{\\Delta_{t}^{3} \\frac{\\partial^{3}{u}}{\\partial{t^{3}}} \\left(7 s_{1} s_{2} - 6 s_{1} - 6 s_{2} + 6\\right)}{6} \\\\ + \\frac{\\Delta_{t}^{2} \\Delta_{x}^{2} \\left(\\frac{\\partial^{4}{C}}{\\partial{x^{2}}\\partial{t^{2}}} s_{1} s_{2} - \\frac{\\partial^{4}{C}}{\\partial{x^{2}}\\partial{t^{2}}} s_{2} + 3 \\frac{\\partial^{4}{\\Lambda}}{\\partial{x^{2}}\\partial{t^{2}}} s_{1} s_{2} - 3 \\frac{\\partial^{4}{\\Lambda}}{\\partial{x^{2}}\\partial{t^{2}}} s_{2} - 3 \\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}} s_{1} - 3 \\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}} s_{2} + 6 \\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}}\\right)}{12} \\\\+ \\frac{\\Delta_{t}^{2} \\Delta_{x} \\frac{\\partial^{3}{B}}{\\partial{x}\\partial{t^{2}}} s_{1} \\left(s_{2} - 1\\right)}{2} + \\frac{\\Delta_{t}^{2} \\frac{\\partial^{2}{u}}{\\partial{t^{2}}} \\left(- 3 s_{1} s_{2} + 2 s_{1} + 2 s_{2}\\right)}{2} - \\frac{\\Delta_{t} \\Delta_{x}^{3} \\frac{\\partial^{4}{B}}{\\partial{x^{3}}\\partial{t}} s_{1} \\left(s_{2} - 1\\right)}{6} + \\frac{\\Delta_{t} \\Delta_{x}^{2} \\left(- \\frac{\\partial^{3}{C}}{\\partial{x^{2}}\\partial{t}} s_{1} s_{2} + \\frac{\\partial^{3}{C}}{\\partial{x^{2}}\\partial{t}} s_{2} - 3 \\frac{\\partial^{3}{\\Lambda}}{\\partial{x^{2}}\\partial{t}} s_{1} s_{2} + 3 \\frac{\\partial^{3}{\\Lambda}}{\\partial{x^{2}}\\partial{t}} s_{2} + 3 \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}} s_{1} + 3 \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}} s_{2} - 6 \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}}\\right)}{6} - \\Delta_{t} \\Delta_{x} \\frac{\\partial^{2}{B}}{\\partial{x}\\partial{t}} s_{1} \\left(s_{2} - 1\\right) \\\\ + \\Delta_{t} \\frac{\\partial{u}}{\\partial{t}} s_{1} s_{2} + \\frac{\\Delta_{x}^{4} s_{2} \\left(\\frac{\\partial^{4}{C}}{\\partial{x^{4}}} s_{1} - 2 \\frac{\\partial^{4}{C}}{\\partial{x^{4}}} + 3 \\frac{\\partial^{4}{\\Lambda}}{\\partial{x^{4}}} s_{1} - 6 \\frac{\\partial^{4}{\\Lambda}}{\\partial{x^{4}}}\\right)}{72} + \\frac{\\Delta_{x}^{3} \\frac{\\partial^{3}{B}}{\\partial{x^{3}}} s_{1} s_{2}}{6} + \\frac{\\Delta_{x}^{2} s_{2} \\left(\\frac{\\partial^{2}{C}}{\\partial{x^{2}}} s_{1} - 2 \\frac{\\partial^{2}{C}}{\\partial{x^{2}}} + 3 \\frac{\\partial^{2}{\\Lambda}}{\\partial{x^{2}}} s_{1} - 6 \\frac{\\partial^{2}{\\Lambda}}{\\partial{x^{2}}}\\right)}{6} + \\Delta_{x} \\frac{\\partial{B}}{\\partial{x}} s_{1} s_{2}\n",
"\\end{array}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "c153ce85-db39-46a4-ba92-e7ec3802a513",
"metadata": {},
"source": [
"isolating $\\frac{\\partial u}{\\partial t}$:\n",
"\n",
"$$\n",
"\\begin{array}{c}\n",
"\\displaystyle\\frac{\\partial u}{\\partial t} + \\frac{\\Delta_{x}}{\\Delta_{t}}\\frac{\\partial{B}}{\\partial{x}} = \\nu\\frac{\\partial^{2}{C}}{\\partial{x^{2}}} + \\Delta_{t}^{3} \\frac{\\partial^{4}{u}}{\\partial{t^{4}}} \\left(\\frac{5}{8} - \\frac{7}{12 s_{2}} - \\frac{7}{12 s_{1}} + \\frac{1}{2 s_{1} s_{2}}\\right) + \\Delta_{t}^{2} \\Delta_{x} \\frac{\\partial^{4}{B}}{\\partial{x}\\partial{t^{3}}} \\left(\\frac{1}{6} - \\frac{1}{6 s_{2}}\\right) + \\Delta_{t}^{2} \\frac{\\partial^{3}{u}}{\\partial{t^{3}}} \\left(- \\frac{7}{6} + \\frac{1}{s_{2}} + \\frac{1}{s_{1}} - \\frac{1}{s_{1} s_{2}}\\right) + \\Delta_{t} \\Delta_{x}^{2} \\frac{\\partial^{4}{C}}{\\partial{x^{2}}\\partial{t^{2}}} \\left(- \\frac{1}{12} + \\frac{1}{12 s_{1}}\\right)\\\\ + \\Delta_{t} \\Delta_{x}^{2} \\frac{\\partial^{4}{\\Lambda}}{\\partial{x^{2}}\\partial{t^{2}}} \\left(- \\frac{1}{4} + \\frac{1}{4 s_{1}}\\right) + \\Delta_{t} \\Delta_{x}^{2} \\frac{\\partial^{4}{u}}{\\partial{x^{2}}\\partial{t^{2}}} \\left(\\frac{1}{4 s_{2}} + \\frac{1}{4 s_{1}} - \\frac{1}{2 s_{1} s_{2}}\\right) + \\Delta_{t} \\Delta_{x} \\frac{\\partial^{3}{B}}{\\partial{x}\\partial{t^{2}}} \\left(- \\frac{1}{2} + \\frac{1}{2 s_{2}}\\right) + \\Delta_{t} \\frac{\\partial^{2}{u}}{\\partial{t^{2}}} \\left(\\frac{3}{2} - \\frac{1}{s_{2}} - \\frac{1}{s_{1}}\\right) + \\Delta_{x}^{3} \\frac{\\partial^{4}{B}}{\\partial{x^{3}}\\partial{t}} \\left(\\frac{1}{6} - \\frac{1}{6 s_{2}}\\right) \\\\+ \\Delta_{x}^{2} \\frac{\\partial^{3}{C}}{\\partial{x^{2}}\\partial{t}} \\left(\\frac{1}{6} - \\frac{1}{6 s_{1}}\\right) + \\Delta_{x}^{2} \\frac{\\partial^{3}{\\Lambda}}{\\partial{x^{2}}\\partial{t}} \\left(\\frac{1}{2} - \\frac{1}{2 s_{1}}\\right) + \\Delta_{x}^{2} \\frac{\\partial^{3}{u}}{\\partial{x^{2}}\\partial{t}} \\left(- \\frac{1}{2 s_{2}} - \\frac{1}{2 s_{1}} + \\frac{1}{s_{1} s_{2}}\\right) + \\Delta_{x} \\frac{\\partial^{2}{B}}{\\partial{x}\\partial{t}} \\left(1 - \\frac{1}{s_{2}}\\right) + \\frac{\\Delta_{x}^{4} \\frac{\\partial^{4}{C}}{\\partial{x^{4}}} \\left(- \\frac{1}{72} + \\frac{1}{36 s_{1}}\\right)}{\\Delta_{t}} + \\frac{\\Delta_{x}^{4} \\frac{\\partial^{4}{\\Lambda}}{\\partial{x^{4}}} \\left(- \\frac{1}{24} + \\frac{1}{12 s_{1}}\\right)}{\\Delta_{t}} - \\frac{\\Delta_{x}^{3} \\frac{\\partial^{3}{B}}{\\partial{x^{3}}}}{6 \\Delta_{t}} + \\frac{\\Delta_{x}^{2} \\frac{\\partial^{2}{\\Lambda}}{\\partial{x^{2}}} \\left(- \\frac{1}{2} + \\frac{1}{s_{1}}\\right)}{\\Delta_{t}}\n",
"\\end{array}\n",
"$$(Full-non-linear-fdm-convec-diff)\n",
"where $\\nu=c_{s}^{2}\\left(\\displaystyle\\frac{2-s1}{2s1}\\right)\\displaystyle\\frac{\\Delta x^{2}}{\\Delta t}$"
]
},
{
"cell_type": "markdown",
"id": "bae98490-7102-445c-9bdc-9d90aa528b90",
"metadata": {},
"source": [
"To correct reduce the above equation and isolate the third and fourth order of spatial derivative terms, we employ the relations:\n",
"\n",
"$$\n",
" \\displaystyle\\frac{\\partial u}{\\partial t} = \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "4543dfd1-6d71-4395-805c-aa55e49c136f",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{3}u}{\\partial x^{2}\\partial t} = \\nu \\displaystyle\\frac{\\partial^{4}C}{\\partial x^{4}} - \\frac{\\partial^{3}B}{\\partial x^{3}}\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "efa1d8bb-02f4-4ad9-b2a7-b945b7dd0e3a",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{2}u}{\\partial t^{2}} = \\nu \\displaystyle\\frac{\\partial^{3}C}{\\partial x^{2}\\partial t} - \\frac{\\partial^{2}B}{\\partial x\\partial t} = \\nu \\displaystyle \\frac{\\partial^{2}}{\\partial x^{2}}\\left(\\frac{\\partial u}{\\partial t}C_{u} \\right) - \\frac{\\partial }{\\partial x}\\left( \\frac{\\partial u}{\\partial t}B_{u} \\right) = \\nu \\displaystyle \\frac{\\partial^{2}}{\\partial x^{2}}\\left[C_{u}\\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right) \\right] - \\frac{\\partial }{\\partial x}\\left[ B_{u}\\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right) \\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "53c88084-02f7-4eaa-8a32-d04d42847d00",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{2}u}{\\partial t^{2}} = \\nu \\displaystyle\\frac{\\partial^{3}C}{\\partial x^{2}\\partial t} - \\frac{\\partial^{2}B}{\\partial x\\partial t} = \\nu \\displaystyle \\frac{\\partial^{2}}{\\partial x^{2}}\\left(\\frac{\\partial u}{\\partial t}C_{u} \\right) - \\frac{\\partial }{\\partial x}\\left( \\frac{\\partial u}{\\partial t}B_{u} \\right) = \\nu \\displaystyle \\frac{\\partial^{2}}{\\partial x^{2}}\\left[C_{u}\\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right) \\right] - \\frac{\\partial }{\\partial x}\\left[ \\nu B_{u} \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial \\Lambda}{\\partial x} \\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "c94b8b3f-a6d0-4f0b-af5d-81199db70db3",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{4}u}{\\partial x^{2}\\partial t^{2}} = \\nu \\displaystyle\\frac{\\partial^{5}C}{\\partial x^{4}\\partial t} - \\frac{\\partial^{4}B}{\\partial x^{3}\\partial t} = \\nu \\displaystyle \\frac{\\partial^{4}}{\\partial x^{4}}\\left(\\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right)C_{u} \\right) - \\frac{\\partial^{3} }{\\partial x^{3}}\\left( \\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right)B_{u} \\right) \n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "7b27b0e7-4970-454a-bc19-f30d9bde3eb5",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{3}u}{\\partial t^{3}} = \\nu \\displaystyle\\frac{\\partial^{4}C}{\\partial x^{2}\\partial t^{2}} - \\frac{\\partial^{3}B}{\\partial x\\partial t^{2}} = \\nu \\frac{\\partial^{2}}{\\partial x^{2}}\\left[ C_{uu} \\left(\\frac{\\partial u}{\\partial t}\\right)^{2} + C_u \\frac{\\partial^{2}u}{\\partial t^{2}} \\right] - \\frac{\\partial}{\\partial x}\\left[B_{uu}\\left(\\frac{\\partial u}{\\partial t}\\right)^{2} + B_u \\frac{\\partial^{2}u}{\\partial t^{2}} \\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "37e12865-978c-4825-b6fa-aa793ec10b84",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{4}u}{\\partial t^{4}} = \\nu \\displaystyle\\frac{\\partial^{5}C}{\\partial x^{2}\\partial t^{3}} - \\frac{\\partial^{4}B}{\\partial x\\partial t^{3}} = \\nu \\frac{\\partial^{2}}{\\partial x^{2}} \\left[ C_{uuu}\\left(\\frac{\\partial u}{\\partial t}\\right)^{3} + 3C_{uu}\\left(\\frac{\\partial u}{\\partial t}\\right)\\left(\\frac{\\partial^{2}u}{\\partial t^{2}}\\right) + C_u \\frac{\\partial^{3}u}{\\partial t^{3}} \\right] - \\frac{\\partial}{\\partial x} \\left[B_{uuu}\\left(\\frac{\\partial u}{\\partial t}\\right)^{3} + 3B_{uu}\\left(\\frac{\\partial u}{\\partial t}\\right)\\left(\\frac{\\partial^{2}u}{\\partial t^{2}}\\right) + B_u \\frac{\\partial^{3}u}{\\partial t^{3}} \\right]\n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "4dc700a5-59a4-4267-8d9e-6c3a8fc3bff8",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial B}{\\partial t} = B_{u}\\frac{\\partial u}{\\partial t} = B_{u}\\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right) = \\nu B_{u} \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - B_{u}^{2}\\frac{\\partial u}{\\partial x} = \\nu B_{u} \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial \\Lambda}{\\partial x} \n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "49a250f4-2cc7-461f-ab24-0a21786e8226",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial^{2} B}{\\partial x \\partial t} = \\frac{\\partial }{\\partial x}\\left(B_{u}\\frac{\\partial u}{\\partial t}\\right) = \\frac{\\partial B_{u}}{\\partial x} \\frac{\\partial u}{\\partial t} + B_{u}\\frac{\\partial u}{\\partial x\\partial t} = \\frac{\\partial B_{u}}{\\partial x} \\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right) + B_{u}\\left( \\nu \\displaystyle\\frac{\\partial^{3}C}{\\partial x^{3}} - \\frac{\\partial^{2} B}{\\partial x^{2}} \\right) = \\frac{\\partial B_{u}}{\\partial x} \\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}}\\right) + B_{u}\\left( \\nu \\displaystyle\\frac{\\partial^{3}C}{\\partial x^{3}} \\right) - \\frac{\\partial^{2} \\Lambda}{\\partial x^{2}} \n",
"$$"
]
},
{
"cell_type": "markdown",
"id": "0a83e5d9-4d1e-4e00-a29e-397cc214a77b",
"metadata": {},
"source": [
"$$\n",
" \\displaystyle\\frac{\\partial C}{\\partial t} = C_{u}\\frac{\\partial u}{\\partial t} = C_{u}\\left( \\nu \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - \\frac{\\partial B}{\\partial x} \\right) = \\nu C_{u} \\displaystyle\\frac{\\partial^{2}C}{\\partial x^{2}} - C_{u}\\frac{\\partial B}{\\partial x}\n",
"$$"
]
}
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