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"---\n",
"title: \"Convergence of FDs for Heat Equation\"\n",
"subtitle: \"Lecture 19\"\n",
"date: 2026-04-20\n",
"abstract-title: Overview\n",
"format:\n",
" html:\n",
" other-links:\n",
" - text: This notebook\n",
" href: L19.ipynb\n",
"---"
]
},
{
"cell_type": "markdown",
"id": "63cf0867",
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"source": [
"::: {.callout-note}\n",
"\n",
"I encourage you to play around with the juptyer notebook for this lecture - you can copy the code with the ```this notebook``` button on the side of this page.\n",
"\n",
":::\n",
"\n",
"::: {.callout-warning}\n",
"\n",
"These notes are mainly a record of what we discussed and are not a substitute for attending the lectures and reading books! If anything is unclear/wrong, let me know and I will update the notes.\n",
"\n",
":::\n",
"\n",
"::: {.callout-tip}\n",
"\n",
"This chapter is mostly based on \n",
"\n",
"* @Burden Chapter 11, \n",
"* @FNC Chapter 10, \n",
"* Lecture 16: @FNC [Galerkin method](https://fncbook.com/galerkin/), \n",
"* Lecture 17: Chapter 14.4 of @Suli: Theorem 14.6. \n",
"* Lecture 19: @Olver2014 Chapter 5\n",
"\n",
":::"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "f8408d86",
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"outputs": [],
"source": [
"#| echo: false\n",
"\n",
"using Plots, LaTeXStrings, LinearAlgebra, Interpolations, SparseArrays\n",
"\n",
"# IF YOU ARE READING THIS, THANK YOU\n",
"# FOR DOWNLOADING THE LECTURE NOTES.\n",
"# THE FIRST PERSON TO LET ME KNOW,\n",
"# WINS SOME CHOCOLATES OF YOUR CHOICE"
]
},
{
"cell_type": "markdown",
"id": "81473756",
"metadata": {},
"source": [
"## Recap\n",
"\n",
"We are now looking at PDEs: \n",
"\n",
"e.g. *Laplace's equation* (Elliptic)\n",
"\n",
"\\begin{align}\n",
" - \\Delta u := \\sum_{i=1}^d \\frac{\\partial^2 u}{\\partial x_i^2} = 0\n",
"\\end{align}\n",
"\n",
"*Wave equation:* (Hyperbolic)\n",
"\n",
"\\begin{align}\n",
" \\square u := u_{tt} - c^2 u_{xx} = 0.\n",
"\\end{align}\n",
"\n",
"The (linear) transport equation ($u_t + c u_{x} = 0$) is a simpler version that displays some of the same behaviour.\n",
"\n",
"*Heat (diffusion) equation* (Parabolic):\n",
"\n",
"\\begin{align}\n",
" u_t - \\alpha^2 u_{xx} = 0.\n",
"\\end{align}\n",
"\n",
"Last time, we saw that $u(x,t) := u_0(x-ct)$ solves the linear transport equation (we used the *method of characteristics*). In A6, you'll implement a finite difference approximation to this problem and look at the so-called CLF condition. \n",
"\n",
"Then, we looked at the heat equation and a forward in time, centred in space (FTCS) finite difference approximation of this problem. Recall that we are looking at a uniform grid with $x_i := a + \\frac{b-a}{m} i$ and $t_j := \\frac{T}{n} j$ and $h := \\frac{b-a}{m}$ and $\\tau := \\frac{T}{n}$. Recall that this approximation can be written as \n",
"\n",
"\\begin{align}\n",
" u_{i,j+1} = (1 - 2\\mu) u_{i,j} + 2\\mu \\frac{u_{i-1,j}+u_{i+1,j}}{2} + f_{ij}\n",
" \\tag{FTCS}\n",
"\\end{align}\n",
"\n",
"where $\\mu := \\tau (\\frac{\\alpha}{h})^2$ is the *diffusion number*. Last time, we saw that, if $0\\leq \\mu \\leq \\frac{1}{2}$, then the method is stable and it is unstable when $\\mu > \\frac{1}{2}$: "
]
},
{
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"id": "7aec19ff",
"metadata": {},
"source": [
"\n",
"::: {#exm-}\n",
"\n",
"Consider the equation\n",
"\n",
"\\begin{align}\n",
" &u_t - \\alpha^2 u_{xx} = 0 \\qquad \\text{on }(0,1)\\nonumber\\\\\n",
" \\tag{IC} &u(x,0) = \\sin \\pi x \\nonumber\\\\\n",
" \\tag{BC} &u(0,t) = u(1,t) = 0.\n",
"\\end{align}\n",
"\n",
"* Show that $u(x,t) = e^{-\\pi^2 t} \\sin \\pi x$ is a solution to this equation. \n",
"\n",
"::: "
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "447761b8",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"\u001b[33m\u001b[1m┌ \u001b[22m\u001b[39m\u001b[33m\u001b[1mWarning: \u001b[22m\u001b[39mμ = 0.5202, unstable!\n",
"\u001b[33m\u001b[1m└ \u001b[22m\u001b[39m\u001b[90m@ Main In[2]:12\u001b[39m\n",
"\u001b[36m\u001b[1m[ \u001b[22m\u001b[39m\u001b[36m\u001b[1mInfo: \u001b[22m\u001b[39mSaved animation to c:\\Users\\math5\\Math 5485\\Math5486\\pics\\heat-3.gif\n"
]
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"text/plain": [
"Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\heat-3.gif\")"
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"source": [
"function fdm_heat( m, n, α, u0; a=0., b=1., f=0., g_a = 0., g_b = 0., T=.2, warn=true )\n",
" # space discretisation\n",
" h = (b-a)/m\n",
" x = collect(a:h:b)\n",
"\n",
" # time discretisation\n",
" τ = T/n \n",
" t = collect(0:τ:T)\n",
"\n",
" μ = τ*(α/h)^2\n",
" if ((μ > 1/2) && (warn))\n",
" @warn \"μ = $μ, unstable!\"\n",
" end\n",
"\n",
" u = zeros( m+1, n+1 )\n",
" u[:,1] = u0.(x)\n",
"\n",
" # boundary condition @ t=0\n",
" u[1,1] = typeof(g_a) <: Function ? g_a(t[1]) : g_a\n",
" u[m+1,1] = typeof(g_b) <: Function ? g_b(t[1]) : g_b\n",
" for j ∈ 1:n\n",
" for i ∈ 2:m\n",
" source = typeof(f) <: Function ? f(x[i],t[i]) : f\n",
" u[i,j+1] = u[i,j] + μ * (u[i-1,j]-2u[i,j]+u[i+1,j]) + τ*source\n",
" end\n",
" u[1,j+1] = typeof(g_a) <: Function ? g_a(t[j+1]) : g_a\n",
" u[m+1,j+1] = typeof(g_b) <: Function ? g_b(t[j+1]) : g_b\n",
" end\n",
"\n",
" return x,t,u\n",
"end\n",
"\n",
"# initial condition\n",
"u0(x) = sin(π*x) \n",
"# exact solution\n",
"u_e(x,t) = exp( -(π^2) * t ) * sin( π*x )\n",
"x,t,u = fdm_heat(50, 1000, 1., u0)\n",
"μ = (.2/1000)*(50)^2\n",
"\n",
"m=51\n",
"x′,t′,u′ = fdm_heat(m, 1000, 1., u0)\n",
"μ′ = (.2/1000)*(m)^2\n",
"\n",
"anim = @animate for j ∈ 1:10:length(t)\n",
" scatter(x, u[:, j]; \n",
" title = L\"Heat equation $u_t - u_{xx} = 0$\",\n",
" label = \"μ = $μ, t = $(t[j])\", \n",
" ylims = [0,1], m=2)\n",
" plot!(x′, u′[:, j]; \n",
" label = \"μ = $(round(μ′,digits=2)), t = $(t[j])\", m=2)\n",
" plot!( x->u_e(x,t[j]); \n",
" label=\"exact solution\", linestyle=:dot)\n",
"end\n",
"gif(anim, \"pics/heat-3.gif\")\n"
]
},
{
"cell_type": "markdown",
"id": "4f77ee08",
"metadata": {},
"source": [
"We also saw that this method with fixed $0 \\leq \\mu \\leq \\frac{1}2$ converges with rate at least $O(h^2)$ when measured in the max norm:"
]
},
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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# | echo: false\n",
"\n",
"α, a, b, T = 1., 0., 1., .2\n",
"\n",
"# compute the error curves for fixed μ = α^2 τ/h^2 \n",
"μμ = [.05, .1, 1/6, .3, .4, .5]\n",
"mm = [2^j for j∈2:8]\n",
"errs = zeros(length(μμ), length(mm))\n",
"for (i,m) ∈ enumerate(mm), (idx,μ) ∈ enumerate(μμ)\n",
" h = (b-a)/m\n",
" n = Int( round( (α^2)*T/(μ*h^2) ) )\n",
" x, t, u = fdm_heat(m, n, 1., u0; warn=false)\n",
" errs[idx,i] = maximum(abs.(u[:, end] - u_e.(x, T)))\n",
"end\n",
"\n",
"plot( mm, errs';\n",
" title=\"error curves\", \n",
" ylims=(minimum(errs[3,:]),1e-1),\n",
" xaxis=:log,yaxis=:log, m=2, xlabel=L\"m\", ylabel=\"error\",\n",
" label=reshape([\"μ = $(round(μ,digits=2))\" for μ ∈ μμ],1,:) )\n",
"plot!( mm, mm.^(-2); linestyle=:dot, label=L\"O(h^2)\")\n",
"plot!( mm, mm.^(-4); linestyle=:dot, label=L\"O(h^4)\")"
]
},
{
"cell_type": "markdown",
"id": "3047e602",
"metadata": {},
"source": [
"Today: We will prove this behaviour.\n",
"\n",
"## Consistency\n",
"\n",
"Again, we apply a Taylor's expansion to the actual solution: \n",
"\n",
"\\begin{align}\n",
" \\frac{u(x,t + \\tau) - u(x,t)}{\\tau} = u_t + \\frac{\\tau}{2} u_{tt} + O(\\tau^2)\n",
"\\end{align}\n",
"\n",
"(where the functions on the right hand side are evaluated at $(x,t)$). Moreover, we have \n",
"\n",
"\\begin{align}\n",
" \\frac{u(x-h,t)-2u(x,t) + u(x+h,t)}{h^2}\n",
" %\n",
" &= u_{xx} + \\frac{h^2}{12} u_{xxxx} + O(h^4). \n",
"\\end{align}\n",
"\n",
"Therefore, we have \n",
"\n",
"\\begin{align}\n",
" &\\frac{u(x,t + \\tau) - u(x,t)}{\\tau} - \\alpha^2 \\frac{u(x-h,t)-2u(x,t) + u(x+h,t)}{h^2} \\nonumber\\\\\n",
" %\n",
" &= u_t - \\alpha^2 u_{xx} + \\frac{\\tau}{2} u_{tt} - \\alpha^2 \\frac{h^2}{12} u_{xxxx}\n",
" %\n",
" + O(\\tau^2) + O(h^4) \\nonumber\\\\\n",
" %\n",
" &= \\frac{\\tau}{2} u_{tt} - \\alpha^2 \\frac{h^2}{12} u_{xxxx}\n",
" %\n",
" + O(\\tau^2) + O(h^4).\n",
"\\end{align}\n",
"\n",
"Therefore, as $h,\\tau\\to 0$, the local trunction error vanishes. That is, the method is consistent with order of accuracy at least $O(\\tau + h^2)$. If one fixes $\\mu = \\tau (\\frac{\\alpha}{h})^2$, then the order of accuracy is at least $O(h^2)$. We also saw (numerically) that, when $\\mu = \\frac{1}{6}$, then the order of accuracy is actually $O(h^4)$:\n",
"\n",
"::: {#exr-}\n",
"\n",
"Show that, if $u$ is smooth, then $u_{tt} = \\alpha^4 u_{xxxx}$. Hint: $(u_{xx})_t = (u_t)_{xx}$. \n",
"\n",
"::: \n",
"\n",
"As a result, we have \n",
"\n",
"\\begin{align}\n",
" &\\frac{\\tau}{2} u_{tt} - \\alpha^2 \\frac{h^2}{12} u_{xxxx} \\nonumber\\\\\n",
" %\n",
" &= \\alpha^2 \\left( \\alpha^2 \\frac{\\tau}{2} - \\frac{h^2}{12} \\right) u_{xxxx} \\nonumber\\\\\n",
" %\n",
" &= \\frac{1}{2} (h \\alpha)^2 \n",
" \\left( \\alpha^2 \\frac{\\tau}{h^2} - \\frac{1}{6} \\right) u_{xxxx}.\n",
"\\end{align}\n",
"\n",
"Therefore, if $\\mu = \\frac{1}{6}$, the leading order in the local trunction error is $O(h^4)$. \n",
"\n",
"## (von Neumann) Stability \n",
"\n",
"We want to explain why our FTCS method converges (when $0 \\leq \\mu \\leq \\frac{1}{2}$). We have shown this method is consistent so we are left with proving stability. We will use the so-called *von Neumann stability analysis*. We actually did this already for the linear advection-diffusion-reaction equations when we looked at the discrete Fourier transform.\n",
"\n",
"The idea is to suppose that, at time $t_j$, the numerical solution is a complex exponential and look at the numerical solution at future times. Because finite differences applied to complex exponentials give constant multiples of complex exponentials (i.e. they are eigenvectors of the corresponding differentiation matrices), we can see when this numerical solution remains bounded as $t \\to \\infty$. \n",
"\n",
"Let $u_{i,j} := e^{\\mathrm{i} k x_i}$. Apply one step of the $\\text{(FTCS)}$ method (here, we take $f = 0$),\n",
"\n",
"\\begin{align}\n",
" u_{i,j+1} &= (1 - 2\\mu) e^{\\mathrm{i} k x_i }+ 2\\mu \\frac{e^{-\\mathrm{i} k h} + e^{\\mathrm{i} k h}}{2} e^{\\mathrm{i} k x_i } \\nonumber\\\\\n",
" %\n",
" &= \\left[1 - 2\\mu + 2\\mu \\cos k h \\right] e^{\\mathrm{i} k x_i } \\nonumber\\\\\n",
" %\n",
" &= \\left[1 - 4\\mu \\sin^2 \\frac{k h}{2} \\right] e^{\\mathrm{i} k x_i } \\nonumber\\\\\n",
" %\n",
" &=: \\lambda(k) u_{i,j}.\n",
"\\end{align}\n",
"\n",
"That is, one step of the method is the same as multiplication by $\\lambda(k)$ and so $u_{i,j+p} = \\lambda(k)^p u_{i,j}$ and the numerical solution remains bounded if and only if $|\\lambda(k)| \\leq 1$. In our case, that is if \n",
"\n",
"\\begin{align}\n",
" & -1 \\leq 1 - 4\\mu \\sin^2 \\frac{k h}{2} \\leq 1 \\nonumber\\\\\n",
" %\n",
" &\\iff 0 \\leq \\mu \\sin^2 \\frac{k h}{2} \\leq \\frac{1}{2}.\n",
"\\end{align}\n",
"\n",
"Therefore, when $0 \\leq \\mu \\leq \\frac{1}{2}$, the method $\\text{(FTCS)}$ is stable. In this case we say the method is *conditionally stable* because we have the restriction on the diffusion number $\\mu := \\tau (\\frac{\\alpha}{h})^2$."
]
},
{
"cell_type": "markdown",
"id": "5835402a",
"metadata": {},
"source": [
"## An implicit method\n",
"\n",
"One may ask if there is an unconditionally stable finite difference approximation to the heat equation. For this, we replace the simple forwards difference with the (implicit) backwards difference. (Recall from A3 that implicit methods can have larger regions of absolute stability, so this approach seems reasonable!)\n",
"\n",
"Consider\n",
"\n",
"\\begin{align}\n",
" u_t(x, t+\\tau) \\approx \\frac{u(x,t+\\tau) - u(x,t)}{\\tau}\n",
"\\end{align}\n",
"\n",
"and so we can approximate $u_t - \\alpha^2 u_{xx} = 0$ with \n",
"\n",
"\\begin{align}\n",
" \\frac{u_{i,j+1} - u_{i,j}}{\\tau} - \\alpha^2 \\frac{u_{i-1,j+1} - 2u_{i,j+1} + u_{i+1,j+1}}{h^2} = 0\n",
"\\end{align}\n",
"\n",
"which is equivalent to \n",
"\n",
"\\begin{align}\n",
" -\\mu u_{i-1,j+1} + (1 + 2\\mu) u_{i,j+1} - \\mu u_{i+1,j+1} = u_{i,j}\n",
" \\tag{BTCS}\n",
"\\end{align}\n",
"\n",
"where $\\mu = \\tau (\\frac{\\alpha}{h})^2$. Notice that this is an implicit equation for $u_{\\bullet,j+1}$. Again, to fix the initial condition, we set $u_{i,0} := u_0(x_i)$ and, to get the correct boundary conditions, we have $u_{0,j} := g_a(t_j)$ and $u_{m,j} := g_b(t_j)$. The internal degrees of freedom are governed by the linear equation \n",
"\n",
"\\begin{align}\n",
" \\begin{pmatrix}\n",
" 1+2\\mu & -\\mu & & \\\\\n",
" -\\mu & 1 + 2\\mu & -\\mu \\\\\n",
" &-\\mu & \\ddots & \\ddots \\\\\n",
" &&\\ddots& \\ddots& -\\mu\\\\\n",
" & & & -\\mu & 1+2\\mu\n",
" \\end{pmatrix}\n",
" u_{\\bullet, j+1}\n",
" %\n",
" = u_{\\bullet, j}\n",
" %\n",
" + \\mu \\begin{pmatrix}\n",
" g_a(t_j) \\\\ 0 \\\\ \\vdots \\\\ 0 \\\\ g_b(t_j)\n",
" \\end{pmatrix}\n",
"\\end{align}\n",
"\n",
"where this matrix is $(m-1)\\times(m-1)$ and $u_{\\bullet,j} := \\{ u_{i,j} \\}_{i=1}^{m-1}$. Solving this linear system of equations gives $u_{\\bullet, j+1}$. We implement this method here: "
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "a6132e15",
"metadata": {},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"\u001b[36m\u001b[1m[ \u001b[22m\u001b[39m\u001b[36m\u001b[1mInfo: \u001b[22m\u001b[39mSaved animation to c:\\Users\\math5\\Math 5485\\Math5486\\pics\\heat-4.gif\n"
]
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"text/plain": [
"Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\heat-4.gif\")"
]
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"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function fdm_heat_backwards( m, n, α, u0; a=0., b=1., g_a = 0., g_b = 0., T=.2, warn=true )\n",
" # space discretisation\n",
" h = (b-a)/m\n",
" x = collect(a:h:b)\n",
"\n",
" # time discretisation\n",
" τ = T/n \n",
" t = collect(0:τ:T)\n",
"\n",
" μ = τ*(α/h)^2\n",
"\n",
" u = zeros( m+1, n+1 )\n",
" u[:,1] = u0.(x)\n",
" \n",
" for j ∈ 1:n\n",
" A = Tridiagonal( fill(-μ,m-2), fill(1+2μ,m-1), fill(-μ,m-2) )\n",
" v = zeros(m-1); \n",
" v[1] = typeof(g_a) <: Function ? g_a(t[j]) : g_a\n",
" v[end] = typeof(g_b) <: Function ? g_b(t[j]) : g_b\n",
"\n",
" u[2:m,j+1] = A \\ ( u[2:m,j] + μ*v ) \n",
" u[1,j+1] = typeof(g_a) <: Function ? g_a(t[j+1]) : g_a\n",
" u[m+1,j+1] = typeof(g_b) <: Function ? g_b(t[j+1]) : g_b\n",
" end\n",
"\n",
" return x,t,u\n",
"end\n",
"\n",
"# initial condition\n",
"u0(x) = sin(π*x) \n",
"# exact solution\n",
"u_e(x,t) = exp( -(π^2) * t ) * sin( π*x )\n",
"x,t,u = fdm_heat_backwards(50, 1000, 1., u0)\n",
"μ = (.2/1000)*(50)^2\n",
"\n",
"m=51\n",
"x′,t′,u′ = fdm_heat_backwards(m, 1000, 1., u0)\n",
"μ′ = (.2/1000)*(m)^2\n",
"\n",
"anim = @animate for j ∈ 1:10:length(t)\n",
" scatter(x, u[:, j]; \n",
" title = L\"Heat equation $u_t - u_{xx} = 0$ (implicit scheme)\",\n",
" label = \"μ = $μ, t = $(t[j])\", \n",
" ylims = [0,1], m=2)\n",
" plot!(x′, u′[:, j]; \n",
" label = \"μ = $(round(μ′,digits=2)), t = $(t[j])\", m=2)\n",
" plot!( x->u_e(x,t[j]); \n",
" label=\"exact solution\", linestyle=:dot)\n",
"end\n",
"gif(anim, \"pics/heat-4.gif\")\n"
]
},
{
"cell_type": "markdown",
"id": "09a132fb",
"metadata": {},
"source": [
"::: {#exr-}\n",
"\n",
"Why is this method computationally feasible dispite having to solve a linear system of equations at each timestep? What is the computational cost to solve the linear system?\n",
"\n",
":::\n",
"\n",
"::: {#exr-}\n",
"\n",
"Complete a von Neumann stability analysis on the method $\\text{(BTCS)}$. Hint: Follow the same procedure as before: Suppose $u_{i,j} := e^{\\mathrm{i} k x_i}$ and $u_{i,j+1} = \\lambda(k) e^{\\mathrm{i} k x_i}$. What is $|\\lambda(k)|$?\n",
"\n",
":::"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "3cbe2fee",
"metadata": {},
"outputs": [
{
"data": {
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",
"image/svg+xml": [
"\n",
"\n"
],
"text/html": [
"
"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# | echo: false\n",
"\n",
"α, a, b, T = 1., 0., 1., .2\n",
"\n",
"# compute the error curves for fixed μ = α^2 τ/h^2 \n",
"μμ = [.05, .1, 1/6, .3, .4, .5]\n",
"mm = [2^j for j∈2:8]\n",
"errs = zeros(length(μμ), length(mm))\n",
"for (i,m) ∈ enumerate(mm), (idx,μ) ∈ enumerate(μμ)\n",
" h = (b-a)/m\n",
" n = Int( round( (α^2)*T/(μ*h^2) ) )\n",
" x, t, u = fdm_heat_backwards(m, n, 1., u0; warn=false)\n",
" errs[idx,i] = maximum(abs.(u[:, end] - u_e.(x, T)))\n",
"end\n",
"\n",
"plot( mm, errs';\n",
" title=\"error curves\", \n",
" ylims=(minimum(errs[3,:]),1e-1),\n",
" xaxis=:log,yaxis=:log, m=2, xlabel=L\"m\", ylabel=\"error\",\n",
" label=reshape([\"μ = $(round(μ,digits=2))\" for μ ∈ μμ],1,:) )\n",
"plot!( mm, mm.^(-2); linestyle=:dot, label=L\"O(h^2)\")\n",
"plot!( mm, mm.^(-4); linestyle=:dot, label=L\"O(h^4)\")"
]
},
{
"cell_type": "markdown",
"id": "dc050a51",
"metadata": {},
"source": [
"::: {#exr-}\n",
"\n",
"Is there a choice of $\\mu$ for which $\\text{(BTCS)}$ is $O(h^4)$ accurate?\n",
"\n",
":::\n",
"\n",
"::: {.box}\n",
"\n",
"Suppose that $u$ is smooth. Since $u(x_i,t_{j}) = u(x_i, t_{j+1} - \\tau) = u(x_i, t_{j+1}) - \\tau u_t(x_i, t_{j+1}) + \\frac{\\tau^2}{2} u_{tt}(x_i, t_{j+1}) + ....$, we have \n",
"\n",
"\\begin{align}\n",
" &\\frac{u(x_i,t_{j+1}) - u(x_i,t_{j})}{\\tau} - \\alpha^2 \\frac{u(x_{i-1},t_{j+1}) - 2u(x_i,t_{j+1}) + u(x_{i+1},t_{j+1})}{h^2} \\nonumber\\\\\n",
" %\n",
" &= \\left( u_t(x_i,t_{j+1}) - \\frac{\\tau}{2} u_{tt}(x_i,t_{j+1}) + O(\\tau^2) \\right) \\nonumber\\\\\n",
" %\n",
" &\\qquad- \\alpha^2 \\left( \n",
" u_{xx}(x_i, t_{j+1}) + \\frac{h^2}{12} u_{xxxx}(x_i,t_{j+1}) + O(h^4)\n",
" \\right)\n",
"\\end{align}\n",
"\n",
"If $u$ solves the equation $u_t - \\alpha^2 u_{xx}=0$, then we have $u_{tt} = (u_t)_t = (\\alpha^2 u_{xx})_t = \\alpha^2 (u_t)_{xx} = \\alpha^4 u_{xxxx}$ and the local trunction error is \n",
"\n",
"\\begin{align}\n",
" &\\left( u_t- \\alpha^2 u_{xx} \\right)\n",
" %\n",
" - \\frac{\\tau}{2} u_{tt} - \\alpha^2 \\frac{h^2}{12} u_{xxxx}\n",
" %\n",
" + O(\\tau^2) + O(h^4) \\nonumber\\\\\n",
" %\n",
" &= \n",
" - \\frac{(\\alpha h)^2}{2} \\left( \\mu + \\frac{1}{6}\\right) u_{xxxx}\n",
" %\n",
" + O(\\tau^2) + O(h^4)\n",
"\\end{align}\n",
"\n",
"evaluated at $(x_i,t_{j+1})$ where $\\tau (\\tfrac{\\alpha}{h})^2$. Now, since $\\mu$ is positive, we can not choose $h, \\tau$ to cancel the leading order contribution. Therefore, unlike in the explicit scheme, we cannot expect faster convergence in general. \n",
"\n",
":::"
]
},
{
"cell_type": "markdown",
"id": "dece9bba",
"metadata": {},
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"::: {#exr-}\n",
"# Crank-Nicolson\n",
"\n",
"Write the methods we discussed today as \n",
"\n",
"\\begin{align}\n",
" \\tag{FTCS} u_{i,j+1} - u_{i,j} &= \\mu( u_{i-1,j} - 2u_{i,j} + u_{i+1,j} ) \\\\\n",
" \\tag{BTCS} u_{i,j+1} - u_{i,j} &= \\mu( u_{i-1,j+1} - 2u_{i,j+1} + u_{i+1,j+1} ).\n",
"\\end{align}\n",
"\n",
"Taking the average of these methods gives the Crank-Nicolson method. Write this method down and show that it is unconditionally stable. What is the order of accuracy of this method?\n",
"\n",
":::"
]
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"Next: \n",
"\n",
"* Numerical algorithms for the wave equation"
]
},
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"id": "b1d5ab0c",
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"source": [
":::{.callout-tip}\n",
"# TO-DO\n",
"\n",
"* Reading for the rest of the course: @FNC 11.1, 11.2, 12.4 \n",
"* Assignment due today!\n",
"* Take a look at the suggestions for [Presentations/Posters/Papers](/Presentations.html) and sign up to present, \n",
"* The last few lectures have been more difficult: Read through the proofs/exercises from the notes and make sure you are up-to-date, \n",
"* (Sign-up to office hours if you would like)\n",
"\n",
"Previous reading:\n",
"\n",
"* Chapter 1 reading:\n",
" * @Burden sections 5.1 - 5.6, \n",
" * @FNC [zero stability](https://fncbook.com/zerostability/) \n",
"* Chapter 2 reading:\n",
" * Recap matrices: sections 7.1 - 7.2 of @Burden \n",
" * Jacobi & Gauss-Seidel: section 7.3 @Burden \n",
" * CG: section 7.6 @Burden\n",
"* Chapter 3 reading: \n",
" * Sections 11.1 - 11.2 of @Burden: shooting methods\n",
" * Lecture 16: @FNC [Galerkin method](https://fncbook.com/galerkin/)\n",
" * Lecture 17: Chapter 14.4 of @Suli: Theorem 14.6. \n",
" * Lecture 19: @Olver2014 Chapter 5\n",
"\n",
":::"
]
}
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