{ "cells": [ { "cell_type": "markdown", "id": "81b1d2f0", "metadata": {}, "source": [ "---\n", "title: \"Transport and Heat Equations\"\n", "subtitle: \"Lecture 18\"\n", "date: 2026-04-15\n", "abstract-title: Overview\n", "format:\n", " html:\n", " other-links:\n", " - text: This notebook\n", " href: L18.ipynb\n", "---" ] }, { "cell_type": "markdown", "id": "63cf0867", "metadata": {}, "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", "\n", ":::" ] }, { "cell_type": "code", "execution_count": 1, "id": "f8408d86", "metadata": {}, "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 have seen how to approximate the solution to general initial value problems: Find $u:[0,T]\\to \\mathbb R$ such that\n", "\n", "\\begin{align}\n", " &u'(t) = f(t, u(t) ) \\nonumber\\\\\n", " &u(0) = u_0.\n", "\\end{align} \n", "\n", "(e.g exponential growth/decay, population models, ....) We saw Euler's method, general one-step methods, Runge-Kutta methods, multistep methods (and their associated error estimates). We then saw linear boundary value problems of the form: find $u : [a,b] \\to \\mathbb R$ such that \n", "\n", "\\begin{align}\n", " -u''(x) &+ p(x) u'(x) + q(x) u(x) = f(x) \\nonumber\\\\\n", " &u(a) = A, \\quad u(b) = B \\tag{Dirichlet} \\\\\n", " \\text{or} \\qquad &u(a) = u(b), \\quad u'(a) = u'(b) \\tag{Periodic}\n", "\\end{align}\n", "\n", "(e.g. semiconductors, pollutants, drug delivery, asset prices, heat transfer in fluids, ...) We saw finite differences and finite elements and some assosciated error estimates. Now, we'll take what we have learnt so far and apply it to Partial Differential Equations (PDEs). " ] }, { "cell_type": "markdown", "id": "22380333", "metadata": {}, "source": [ "## Some examples...\n", "\n", "PDEs can model a variety of phenomena. Here, we list some examples to introduce some of the notation:\n", "\n", "*Laplace's equation*: \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", "Here, $\\Delta$ is the *Laplacian*. Solutions to this equation are called *Harmonic*. Poisson's equation is the inhomogenous version: $- \\Delta u = f$. This models heat conduction, elastic deformation, appears in electrostatics, Newton's law of gravitation, ....\n", "\n", "*Helmholtz's equation*: \n", "\n", "\\begin{align}\n", " - \\Delta u = \\lambda u\n", "\\end{align}\n", "\n", "This is an eigenvalue equation which appears in the study of acoustics, optics, electromagnetism, it's related to time-independent Schr\\\"odinger equation, .... \n", "\n", "*Heat (diffusion) equation*:\n", "\n", "\\begin{align}\n", " u_t - \\Delta u = 0\n", "\\end{align}\n", "\n", "where $u_t := \\frac{\\partial u}{\\partial t}$. Models the dissipation of some quantity as a function of time. (Related to the Black-Scholes equation in finance).\n", "\n", "*Schrödinger equation:*\n", "\n", "\\begin{align}\n", " i \\hbar \\frac{\\partial \\Psi}{\\partial t} = \\hat{H} \\Psi\n", "\\end{align}\n", "\n", "where $\\hat{H}$ is the *Hamiltonian* and $\\Psi$ is the *wave function* representing the distribution of electrons in a quantum system. \n", "\n", "*Wave equation:*\n", "\n", "\\begin{align}\n", " u_{tt} - \\Delta u = 0.\n", "\\end{align}\n", "\n", "So far, we have only seen linear equations (the equations are linear in $u$ and it's derivatives). Nonlinear equations are also of importance in many fields:\n", "\n", "*Minimal surfaces:*\n", "\n", "\\begin{align}\n", " \\nabla \\cdot \\left( \\frac{\\nabla u}{\\sqrt{1 + |\\nabla u|^2}} \\right) = 0.\n", "\\end{align}\n", "\n", "This equation describes surfaces with zero mean curvature - this is the shape of soap films stretched across a wire boundary. Here, $\\nabla \\cdot v := \\sum_{i=1}^d \\frac{\\partial v_{i}}{\\partial x_i}$ is the *divergence* of $v$. \n", "\n", "*Monge-Ampère equation:*\n", "\n", "\\begin{align}\n", " \\det D^2 u = f.\n", "\\end{align}\n", "\n", "Here, $D^2 u := (u_{x_i x_j})_{i,j=1}^n$ denotes the *Hessian*. This equation arises in optimal transport (moving mass efficiently), differential geometry (prescribing the Gauss curvature of a surface), and meteorology.\n", "\n", "*Burgers':*\n", "\n", "\\begin{align}\n", " u_t + (u \\cdot \\nabla) u = \\nu \\Delta u\n", "\\end{align}\n", "\n", "($\\nu=0$: inviscid Burgers')\n", "\n", "*Korteweg-de Vries (KdV):*\n", "\n", "\\begin{align}\n", " u_t + 6 u u_x + u_{xxx} = 0\n", "\\end{align}\n", "\n", "Waves in shallow water.\n", "\n", "In this course, we will look at *linear PDEs* which can be categorised as Elliptic, Parabolic, or Hyperbolic: consider the general linear equation \n", "\n", "\\begin{align}\n", " A u_{\\xi\\xi} + B u_{\\xi\\zeta} + C u_{\\zeta\\zeta} + \\mathrm{l.o.t} =0\n", "\\end{align}\n", "\n", "where $\\mathrm{l.o.t}$ represents \"lower order terms\" (functions involving derivatives of lower order). Here, we are thinking of either $(\\xi,\\zeta) = (x,t)$ (one spatial dimension) or $(\\xi, \\zeta) = (x,y)$ (two spatial dimensions and no time dependence). To this equation, we may assign the conic section $A \\xi^2 + B \\xi \\zeta + C \\zeta^2 = 0$. The equation is classed as Elliptic, Parabolic, or Hyperbolic if the corresponding conic section is Elliptic, Parabolic, or Hyperbolic, respectively. That is, \n", "\n", "\\begin{align}\n", " &B^2 - 4AC < 0 &&\\text{Elliptic} \\\\\n", " &B^2 - 4AC = 0 &&\\text{Parabolic} \\\\\n", " &B^2 - 4AC > 0 &&\\text{Hyperbolic}\n", "\\end{align}\n", "\n", "Roughly speaking, Elliptic equations describe steady-state systems, Parabolic equations describe diffusion (systems that evolve towards equilibrium), and Hyperbolic equations describe wave propagation. \n", "\n", "::: {#exr-}\n", "\n", "Show that Laplace's equation (in 2d) is Elliptic, the wave equation is Hyperbolic, and the heat equation is Parabolic\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "ee16dc71", "metadata": {}, "source": [ "## Laplace equation (Elliptic)\n", "\n", "See presentation on finite differences for Laplace equation in 2d." ] }, { "cell_type": "markdown", "id": "9c51387e", "metadata": {}, "source": [ "## Transport/advection equation (Hyperbolic)\n", "\n", "The simplest possible equation of hyperbolic type is the linear transport equation with constant coefficient:\n", "\n", "\\begin{align}\n", " & u_t + b \\cdot \\nabla u = 0 \\nonumber\\\\\n", " & u(x,0) = u_0(x).\n", "\\end{align}\n", "\n", "::: {.callout-note}\n", "\n", "We use the notation $\\nabla u$ for the vector of first derivatives with respect to the spatial dimensions: $(\\nabla u)_i = u_{x_i} = \\frac{\\partial u}{\\partial x_i} = \\partial_{x_i} u = \\partial_i u = \\cdots$. The dot is the Euclidean dot product so that $b \\cdot \\nabla u := \\sum_{i=1}^d b_i u_{x_i}$.\n", "\n", ":::\n", "\n", "::: {#exr-}\n", "\n", "Show that $u(x,t) := u_0(x - bt)$ is a solution.\n", "\n", "::: \n", "\n", "***Answer.*** You can show this directly but here we look at a more general method.\n", "\n", "We will use the so-called *method of characteristics* - the idea is to reduce the PDE to a system of ODEs by trying to parametrise the solution as $z(s) := u\\big( x(s), t(s) \\big)$ where $x(0) = \\xi$ and $t(0) = 0$. Then, $z$ satisfies the ODE:\n", "\n", "\\begin{align}\n", " z(0) &= u( \\xi, 0 ) = u_0(\\xi) \\nonumber\\\\\n", " z'(s) &= t'(s) u_t\\big( x(s), t(s) \\big) + x'(s) \\cdot \\nabla u\\big(x(s),t(s)\\big)\n", "\\end{align} \n", "\n", "Here, we have applied the chain rule. Therefore, if \n", "\n", "\\begin{align}\n", " &t(0) = 0, &&t'(s) = 1 \\nonumber\\\\\n", " &x(0) = \\xi &&x'(s) = b\n", "\\end{align}\n", "\n", "(these are known as the *characteristic equations*), then $z'(s) = 0$ and $z(s)$ is constant (and equal to $z(0) = u_0(\\xi)$). The solution to the characteristic equations are given by $x(s) = \\xi + sb$ and $t(s) = s$, and so $z(s) = u(\\xi + sb, s) = u_0(\\xi) = z(0)$ and thus $u(x,t) = u_0(x-tb)$. $\\square$\n", "\n", "We have also shown that for the linear transport equation, the solution is constant along the characteristic curves $(\\xi+sb, s)$:" ] }, { "cell_type": "code", "execution_count": 2, "id": "fcfa9e07", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# | echo: false\n", "\n", "b = 1.\n", "P = plot(xlabel=L\"x\", ylabel=L\"t\", \n", " title=\"characteristic curves for linear transport equation\")\n", "for ξ ∈ -1:.1:1 \n", " scatter!( [ξ], [0], primary=false )\n", " plot!( ξ.+b.*[0,1], [0,1], primary=false, xlims=(-1,1))\n", "end\n", "P" ] }, { "cell_type": "markdown", "id": "5170226b", "metadata": {}, "source": [ "Here, we look at what these solutions do: " ] }, { "cell_type": "code", "execution_count": 3, "id": "b3e12af3", "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\\transport-1.gif\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\transport-1.gif\")" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# | echo: false\n", " \n", "u0(x) = exp(-x^2)\n", "τ=0.1\n", "t = 0:τ:10\n", "\n", "anim = @animate for j ∈ 1:length(t)\n", " b = -1.\n", " plot( x->u0(x-b*t[j]); \n", " ylims=[0,1],\n", " legend_title=\"t = $(t[j])\", legend=:topleft,\n", " title = L\"Transport equation $u_t + b u_{x} = 0$\",\n", " label = \"b = $b\")\n", " b = 0.\n", " plot!( x->u0(x-b*t[j]); label = \"b = $b\")\n", " b = 1.\n", " plot!( x->u0(x-b*t[j]); label = \"b = $b\")\n", "end\n", "gif(anim, \"pics/transport-1.gif\")\n" ] }, { "cell_type": "markdown", "id": "ec91134f", "metadata": {}, "source": [ "Inhomogeneous equation: \n", "\n", "\\begin{align}\n", " & u_t + b \\cdot \\nabla u = f \\nonumber\\\\\n", " & u(x,0) = u_0(x)\n", "\\end{align}\n", "\n", "::: {#exr-}\n", "\n", "Write down an integral form of the solution to this equation. \n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "dd74055c", "metadata": {}, "source": [ "Transport with decay: take $\\mu > 0$ and consider\n", "\n", "\\begin{align}\n", " &u_t + b \\cdot \\nabla u + \\mu u = 0\\nonumber\\\\\n", " &u(x,0) = u_0(x)\n", "\\end{align}\n", "\n", "::: {#exr-}\n", "\n", "Applying the exact same argument as before, we find that $z'(s) + a z(s) = 0$ (where $z(s) = u(\\xi + sb, s)$). Use the integrating factor to conclude that the exact solution is \n", "\n", "\\begin{align}\n", " u(x,t) = u_0(x-bt) e^{-\\mu t}\n", "\\end{align}" ] }, { "cell_type": "code", "execution_count": 4, "id": "43f71d3d", "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\\transport-2.gif\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\transport-2.gif\")" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# | echo: false\n", "\n", "μ = .25\n", "u0(x) = exp(-x^2)\n", "τ=0.1\n", "t = 0:τ:10\n", "\n", "anim = @animate for j ∈ 1:length(t)\n", " b = -1.\n", " plot( x->u0(x-b*t[j])*exp(-μ*t[j]); \n", " ylims=[0,1],\n", " legend_title=\"t = $(t[j])\", legend=:topleft,\n", " title = L\"Transport equation $u_t + b u_{x} + μ u = 0$\",\n", " label = \"b = $b\")\n", " b = 0.\n", " plot!( x->u0(x-b*t[j])*exp(-μ*t[j]); label = \"b = $b\")\n", " b = 1.\n", " plot!( x->u0(x-b*t[j])*exp(-μ*t[j]); label = \"b = $b\")\n", "end\n", "gif(anim, \"pics/transport-2.gif\")\n" ] }, { "cell_type": "markdown", "id": "837ed44d", "metadata": {}, "source": [ ":::" ] }, { "cell_type": "markdown", "id": "7aec19ff", "metadata": {}, "source": [ "## Heat equation (Parabolic)\n", "\n", "The *heat equation* (or the *diffusion equation*) is \n", "\n", "\\begin{align}\n", " u_t - \\alpha^2 \\Delta u = f\n", "\\end{align}\n", "\n", "This models e.g. how temperature density $u$ distributes (diffuses) accross a domain $\\Omega \\subset \\mathbb R^d$. Here, $\\alpha^2$ is the thermal diffusivity. \n", "\n", "::: {#exr-}\n", "\n", "Take $d = 1$. Explain why the term $-\\alpha^2 u''$ makes the solution \"smooth out\" (diffuses) over time. \n", "\n", ":::\n", "\n", "::: {#exr-}\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", "::: \n", "\n", "Here, we look at a finite difference implementation of this equation:" ] }, { "cell_type": "code", "execution_count": 5, "id": "447761b8", "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-1.gif\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\heat-1.gif\")" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "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", " c = τ*(α/h)^2\n", " if ((c > 1/2) && (warn))\n", " @warn \"c = $c, 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[j]) : f\n", " u[i,j+1] = u[i,j] + c * (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", "\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 = \"t = $(t[j])\", \n", " ylims = [0,1], m=2)\n", " plot!( x->u_e(x,t[j]); \n", " label=\"exact solution\", linestyle=:dot)\n", "end\n", "gif(anim, \"pics/heat-1.gif\")\n" ] }, { "cell_type": "markdown", "id": "4dedc99b", "metadata": {}, "source": [ "Here, we run the same code with different mesh sizes:" ] }, { "cell_type": "code", "execution_count": 6, "id": "4315e298", "metadata": {}, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "\u001b[33m\u001b[1m┌ \u001b[22m\u001b[39m\u001b[33m\u001b[1mWarning: \u001b[22m\u001b[39mc = 0.7200000000000001, unstable!\n", "\u001b[33m\u001b[1m└ \u001b[22m\u001b[39m\u001b[90m@ Main In[5]: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-2.gif\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\heat-2.gif\")" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "x,t,u = fdm_heat(60, 1000, 1., u0)\n", "\n", "anim = @animate for j ∈ 1:5:length(t)\n", " plot(x, u[:, j]; \n", " title = L\"Heat equation $u_t - u_{xx} = 0$\",\n", " label = \"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-2.gif\", fps=3)" ] }, { "cell_type": "markdown", "id": "0b2d0b25", "metadata": {}, "source": [ "::: {#exr-}\n", "\n", "* What do you think is happening here?\n", "\n", "Hint: you can rewrite the update step as \n", "\n", "\\begin{align}\n", " u_{i,j+1} &= u_{ij} + c \\big(u_{i-1,j}-2u_{ij}+u_{i+1,j}\\big) \\nonumber\\\\\n", " %\n", " &= (1-2c) u_{ij} + c \\big(u_{i-1,j}+u_{i+1,j}\\big)\n", "\\end{align}\n", "\n", "where $c = \\alpha^2 \\frac{\\tau}{h^2}$.\n", "\n", "* For what $c$ do you think this method is stable? \n", "* For what $c$ do you think this method is convergent? \n", "\n", ":::\n", "\n", "Here, we plot error curves as a function of $h$ with fixed $c = \\alpha^2 \\frac{\\tau}{h^2}$:" ] }, { "cell_type": "code", "execution_count": 7, "id": "74420363", "metadata": {}, "outputs": [ { "data": { "image/png": "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", "image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 7, "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 c = α^2 τ/h^2 \n", "cc = [.05, .1, 1/6, .3, .4, .5]\n", "mm = [2^j for j∈2:8]\n", "errs = zeros(length(cc), length(mm))\n", "for (i,m) ∈ enumerate(mm), (idx,c) ∈ enumerate(cc)\n", " h = (b-a)/m\n", " n = Int( round( (α^2)*T/(c*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([\"c = $(round(c,digits=2))\" for c ∈ cc],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": "1e835048", "metadata": {}, "source": [ "::: {#exr-}\n", "\n", "How would you update the code to include non-zero boundary conditions and a source term $f$:\n", "\n", "\\begin{align}\n", " &u_t - \\alpha^2 u_{xx} = f \\qquad \\text{on }(a,b)\\nonumber\\\\\n", " &u(x,0) = u_0(x) \\nonumber\\\\\n", " &u(a,t) = g_a(t), \\nonumber\\\\\n", " &u(b,t) = g_b(t).\n", "\\end{align}\n", "\n", ":::\n", "\n", "Here, we try and solve the equation on $[-1,1]$ with $g_a = 0$ and $g_b = 2$:" ] }, { "cell_type": "code", "execution_count": 8, "id": "2124d8b7", "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-3.gif\n" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\heat-3.gif\")" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a, b = -1, 1\n", "# initial condition\n", "u0(x) = 1 + sin(π*x/2) + 3 * (1-x^2) * exp(-4x^2);\n", "# boundary\n", "g_a(t) = 0.\n", "g_b(t) = 2.\n", "\n", "x,t,u = fdm_heat(50, 2000, 1., u0; a, b, g_a, g_b, T=1)\n", "\n", "anim = @animate for j ∈ 1:10:length(t)\n", " plot(x, u[:, j]; \n", " title = \"Heat equation (non-zero boundary)\",\n", " label = \"t = $(t[j])\", \n", " ylims = [0,4], m=2)\n", "end\n", "gif(anim, \"pics/heat-3.gif\")\n" ] }, { "cell_type": "markdown", "id": "183d1311", "metadata": {}, "source": [ "Here, we also add a source term:" ] }, { "cell_type": "code", "execution_count": 9, "id": "ee6954d8", "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" ] }, { "data": { "text/html": [ "" ], "text/plain": [ "Plots.AnimatedGif(\"c:\\\\Users\\\\math5\\\\Math 5485\\\\Math5486\\\\pics\\\\heat-4.gif\")" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a, b = -1, 1\n", "# initial condition\n", "u0(x) = 1 + sin(π*x/2) + 3 * (1-x^2) * exp(-4x^2);\n", "# boundary\n", "g_a(t) = 0.\n", "g_b(t) = 2.\n", "# source\n", "f(x,t) = -50x\n", "\n", "x,t,u = fdm_heat(50, 1000, 1., u0; a, b, f, g_a, g_b)\n", "\n", "anim = @animate for j ∈ 1:10:length(t)\n", " plot(x, u[:, j]; \n", " title = \"Heat equation (with source)\",\n", " label = \"t = $(t[j])\", \n", " ylims = [0,4], m=2)\n", "end\n", "gif(anim, \"pics/heat-4.gif\")" ] }, { "cell_type": "markdown", "id": "c79f3e8c", "metadata": {}, "source": [ "Next: \n", "\n", "* We prove convergence of this finite difference scheme for solving the heat equation by showing *(i)* Consistency and *(ii)* (von Neumann) stability criterion,\n", "* Implicit schemes and Crank-Nicolson,\n", "* Periodic boundary conditions and semidiscretisation (\"method of lines\"),\n" ] }, { "cell_type": "markdown", "id": "b1d5ab0c", "metadata": {}, "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", "\n", ":::" ] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.11", "language": "julia", "name": "julia-1.11" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.11.6" } }, "nbformat": 4, "nbformat_minor": 5 }