{ "cells": [ { "cell_type": "markdown", "id": "3d0a47e3-7c72-43fc-8563-d44cf886a24b", "metadata": {}, "source": [ "---\n", "title: \"A3: Absolute Stability\"\n", "subtitle: \"Assignment 3: Absolute stability of Runge-Kutta and multistep methods\"\n", "date: 2026-02-16\n", "format:\n", " html:\n", " other-links:\n", " - text: This notebook\n", " href: A3.ipynb\n", "---\n", "\n", ":::{.callout-note}\n", "\n", "* Complete the following and submit to Canvas before Feb 25, 23:59\n", "* Late work will recieve 0%,\n", "* Each assignment is worth the same, \n", "* Please get in contact with the TA/instructor in plenty of time if you need help,\n", "* Before submitting your work, make sure to check everything runs as expected. Click **Kernel > Restart Kernel and Run All Cells**.\n", "* Feel free to add more cells to experiment or test your answers,\n", "* I encourage you to discuss the course material and assignment questions with your classmates. However, unless otherwise explicitly stated on the assignment, you must complete and write up your solutions on your own,\n", "* The use of GenAI is prohibited as outlined in the course syllabus. If I suspect you of cheating, you may be asked to complete a written or oral exam on the content of this assignment.\n", "\n", ":::" ] }, { "cell_type": "code", "execution_count": null, "id": "5b86f5a6", "metadata": {}, "outputs": [], "source": [ "#| echo: false\n", "\n", "# The following code will be useful\n", "using Plots, LaTeXStrings\n", "\n", "function RK( u0, f, T, n, A, b, c )\n", " h, t = T/n, 0:T/n:T \n", " u = fill(float(u0), n+1)\n", " s, = size(A)\n", " s = s+1 \n", " k = fill(float(u0), s)\n", " for j = 1:n\n", " k[1] = h * f( t[j], u[j] )\n", " for i = 1:s-1 \n", " k[i+1] = h * f( t[j] + h * c[i], u[j] + sum( A[i,ℓ] * k[ℓ] for ℓ ∈ 1:i ) )\n", " end\n", " u[j+1] = u[j] + sum( b[ℓ] * k[ℓ] for ℓ ∈ 1:s )\n", " end\n", " return u\n", "end \n", "\n", "function explicitLinearMultistep( u0, f, T, n, α, β)\n", " m, h, t = length(α), T/n, 0:T/n:T \n", " u = fill(float(u0), n+1)\n", "\n", " # initiate u_1,...,u_m-1 using RK4\n", " if (m > 1)\n", " u[1:m] = RK( u0, f, (m-1)*h, m-1, \n", " [1/2 0 0 ; 0 1/2 0 ; 0 0 1],\n", " [1/6 1/3 1/3 1/6], [1/2 1/2 1] ) \n", " end\n", "\n", " f_hist = [ f(t[m-ℓ],u[m-ℓ]) for ℓ ∈ 1:m-1 ]\n", " for j = m:n\n", " f_hist = [ f(t[j],u[j]), f_hist[1:m-1]... ]\n", " u[j+1] = sum( α[m-ℓ] * u[j-ℓ] + h * β[m-ℓ] * f_hist[ℓ+1] for ℓ ∈ 0:(m-1) )\n", " end \n", " \n", " return u\n", "end " ] }, { "cell_type": "markdown", "id": "93f519cf", "metadata": {}, "source": [ "::: {.box}\n", "\n", "***Name:***\n", "\n", "***Approx. time spent on this assignment:***\n", "\n", "***Did you do the suggested reading?*** Yes/No\n", "\n", ":::\n", "\n", "Section 5.11 of @Burden may be useful.\n", "\n", "In lectures, we have discussed *zero-stability* of numerical methods for solving IVPs. We have seen that Runge-Kutta methods and linear multistep methods that satisfy the root condition are zero stable. Therefore, these methods are convergent: for small enough step size the numerical approximation is \"close\" to the exact solution. This notion does **not** take into account how small the step size has to be in order to achieve a reasonable error. Absolute stability on the other hand is about fixed step size. Roughly speaking, a numerical scheme is absolutely stable for a given step size if $u_j$ remains bounded as $j\\to\\infty$. \n", "\n", "To introduce this concept, we need the following test problem: for $\\lambda \\in \\mathbb C$,\n", "\n", "\\begin{align}\n", " u(0) &= u_0 \\nonumber\\\\\n", " u'(t) &= \\lambda u(t).\n", " \\tag{test}\n", "\\end{align}\n", "\n", "Notice that the exact solution to this equation is $u(t) = u_0 e^{\\lambda t}$ and so, if $\\lambda = x + i y$, then $u(t) = u_0 e^{x t} \\big[ \\cos y + i \\sin y \\big]$. This solution remains bounded if $x \\leq 0$ and diverges when $x > 0$. It turns out that analysing how well numerical schemes can approximate this problem is informative: \n", "\n", "::: {#def-A}\n", "\n", "We say that a numerical scheme is *absolutely stable* for a given $\\lambda \\in \\mathbb C$ and step size $h$ if the numerical approximation, $u_j$, to $\\text{(test)}$ remains bounded as $j \\to \\infty$.\n", "\n", ":::\n", "\n", "::: {#def-R}\n", "\n", "The *region of absolute stability*, $\\mathscr A$, of a numerical method is defined as the set of all $z := h\\lambda$ for which the method is absolutely stable for $\\lambda$ and step size $h$.\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "2cb7d3a8", "metadata": {}, "source": [ "## A. RK Methods " ] }, { "cell_type": "markdown", "id": "757f8cb4", "metadata": {}, "source": [ "**Exercise 1.** Show that Euler's method has the following region of absolute stability\n", "\n", "\\begin{align}\n", " \\mathscr A = \\{ z \\in \\mathbb C : |z + 1| \\leq 1 \\}. \\nonumber\n", "\\end{align}\n", "\n", "(That is, $\\mathscr A$ is a ball of radius $1$ centered at $-1$)\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "6bf9a09e", "metadata": {}, "source": [ "**Exercise 2.** Suppose you apply Euler's method to\n", "\n", "\\begin{align}\n", " u(0) &= 1 \\nonumber\\\\\n", " u'(t) &= -10 u(t). \\nonumber\n", "\\end{align}\n", "\n", "What step size $h$ should you use to ensure the numerical solution remains bounded for large times?\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "2ef378f7", "metadata": {}, "source": [ "**Exercise 3.** Consider your favourite 2 stage, order 2 Runge-Kutta method. Show that the region of absolute stability is given by \n", "\n", "\\begin{align}\n", " \\mathscr A = \\left\\{ z \\in \\mathbb C : \\big| 1 + z + \\tfrac{z^2}{2} \\big| \\leq 1 \\right\\}. \\nonumber\n", "\\end{align}\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "6707e69b", "metadata": {}, "source": [ "It turns out that there exists a *stability function* $R$ such that \n", "\n", "\\begin{align}\n", " \\mathscr A = \\{ z\\in \\mathbb C : |R(z)| \\leq 1 \\}.\\nonumber\n", "\\end{align}\n", "\n", "Notice that for Euler's method $R(z) = 1 + z$ and all 2 stage order 2 RK methods have $R(z) = 1 + z + \\tfrac{z^2}{2}$. In general, for an $s$-stage RK method $R(z)$ is a polynomial of degree $s$. If the method has order $p$, then this polynomial must match the first $p+1$ terms of the Taylor expansion of $e^z$. We plot the region of absolute stabily for the first few RK methods:" ] }, { "cell_type": "code", "execution_count": 18, "id": "2ab9c717", "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" ], "text/html": [ "" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#| echo: false\n", "\n", "x = range(-3.,3.,200)\n", "y = range(-3.,3,200)\n", "\n", "# Euler \n", "R1(x,y) = abs( x+1im*y+1 )\n", "\n", "# 2 stage, order 2\n", "R2(x,y) = abs( 1 + x+1im*y + (x+1im*y)^2/2 )\n", "\n", "# RK4\n", "R3(x,y) = abs( 1 + x+1im*y + (x+1im*y)^2/2 + (x+1im*y)^3/6 + (x+1im*y)^4/24 )\n", "\n", "z1 = [R1(x,y) <= 1.0 ? 1.0 : NaN for y∈y, x∈x]\n", "z2 = [R2(x,y) <= 1.0 ? 1.0 : NaN for y∈y, x∈x]\n", "z3 = [R3(x,y) <= 1.0 ? 1.0 : NaN for y∈y, x∈x]\n", "\n", "heatmap(x, y, z3, \n", " c = :lightblue, \n", " title = \"Region of absolute stability |R(z)| ≤ 1\",\n", " xlabel = \"Re(z)\", ylabel = \"Im(z)\",\n", " colorbar = false, aspect_ratio = :equal)\n", "contour!(x, y, R3.(x',y), \n", " levels=[1], color=:black, lw=2)\n", "\n", "heatmap!(x, y, z2, c = :blue)\n", "contour!(x, y, R2.(x',y),\n", " levels=[1], color=:black, lw=2)\n", "\n", "heatmap!(x, y, z1, c = :darkblue)\n", "contour!(x, y, R1.(x',y),\n", " levels=[1], color=:black, lw=2)\n", "\n", "scatter!([],[], fillalpha=.5, legend=:topright, color=:darkblue, label=\"Euler\")\n", "scatter!([],[], fillalpha=.5, color=:blue, label=\"RK2\")\n", "scatter!([],[], fillalpha=.5, color=:lightblue, label=\"RK4\")" ] }, { "cell_type": "markdown", "id": "a3b5ed19", "metadata": {}, "source": [ "## B. Multistep methods" ] }, { "cell_type": "markdown", "id": "d0b4a3a1", "metadata": {}, "source": [ "Recall that, for a general multistep method\n", "\n", "\\begin{align}\n", " u_{j+1} &= \\alpha_{m-1} u_j + \\cdots + \\alpha_0 u_{j-m+1} \\nonumber\\\\\n", " %\n", " &\\,\\, + h \\Big[ \\beta_m f(t_{j+1},u_{j+1}) + \\beta_{m-1} f(t_j,u_j) + \\cdots + \\beta_0 f(t_{j-m+1},u_{j-m+1}) \\Big], \n", "\\end{align}\n", "\n", "we define the generating polynomials\n", "\n", "\\begin{align}\n", " \\rho(z) &= z^m - \\alpha_{m-1} z^{m-1} - \\cdots - \\alpha_0 \\nonumber\\\\\n", " \\sigma(z) &= \\beta_m z^m + \\beta_{m-1}z^{m-1} + \\dots + \\beta_0. \\nonumber\n", "\\end{align}\n", "\n", "**Exercise 4.** Suppose that $\\zeta$ is a solution of $\\rho(\\zeta) = h\\lambda \\sigma(\\zeta)$ and show that $u_{j} := \\zeta^j$ solves the difference equation $(1)$. \n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::\n", "\n", "It turns out that if all the roots $\\zeta_1,\\dots,\\zeta_m$ of $\\rho - h\\lambda\\sigma$ are simple, then we can write a general solution to $(1)$ as\n", "\n", "\\begin{align}\n", " u_j = c_1 (\\zeta_1)^j + \\cdots + c_m (\\zeta_m)^j \\nonumber\n", "\\end{align}\n", "\n", "for some constants $c_1,\\dots,c_m$. This numerical solution remains bounded if and only if $|\\zeta_j| \\leq 1$ for all $j$." ] }, { "cell_type": "markdown", "id": "364d3c4f", "metadata": {}, "source": [ "**Exercise 5.** Explain why $(\\text{AB2})$ has a bounded region of absolute stability.\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::\n", "\n", "It turns out that all explicit multistep methods have bounded regions of absolute stability. We now plot regions of absolute stability for $(\\text{AB1})$-$(\\text{AB4})$:" ] }, { "cell_type": "code", "execution_count": 9, "id": "4fd91e31", "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" ], "text/html": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#| echo: false\n", "\n", "z = @. exp( 1im * (0:.01:2π) )\n", "\n", "# Euler\n", "ρ1(z) = z - 1 \n", "σ1(z) = 1 \n", "\n", "P = plot( @. real(ρ1(z)/σ1(z)), @. imag(ρ1(z)/σ1(z));\n", " label = \"AB1 (Euler)\", lw=2, fillrange=0, fillcolor=:darkblue,fillalpha=.5, color=:black,\n", " xlabel = \"Re(z)\", ylabel = \"Im(z)\", aspect_ratio = :equal)\n", "\n", "# AB2\n", "ρ1(z) = z^2 - z \n", "σ1(z) = (3/2) * z - 1/2 \n", "\n", "plot!( @. real(ρ1(z)/σ1(z)), @. imag(ρ1(z)/σ1(z));\n", " label = \"AB2\", lw=2, fillrange=0, fillcolor=:blue, fillalpha=.5,color=:black)\n", "\n", "# AB3\n", "ρ1(z) = z^3 - z^2 \n", "σ1(z) = (1/12) * (23z^2 - 16z + 5) \n", "\n", "plot!( @. real(ρ1(z)/σ1(z)), @. imag(ρ1(z)/σ1(z));\n", " label = \"AB3\", lw=2, fillrange=0, fillcolor=:green, fillalpha=.5,color=:black)\n", "\n", "# AB4\n", "ρ1(z) = z^4 - z^3 \n", "σ1(z) = (1/24) * (55z^3 - 59z^2 + 37z - 9) \n", "\n", "plot!( @. real(ρ1(z)/σ1(z)), @. imag(ρ1(z)/σ1(z));\n", " label = \"AB4\", lw=2, fillrange=0, fillcolor=:lightblue, fillalpha=.5, color=:black)" ] }, { "cell_type": "markdown", "id": "0677693f", "metadata": {}, "source": [ "Here, we plot the errors when approximating \n", "\n", "\\begin{align}\n", " u(0) &= 1 \\nonumber\\\\\n", " u'(t) &= -10 u(t)\n", "\\end{align}\n", "\n", "on $[0,100]$ for the methods $(\\text{AB1})-(\\text{AB4})$" ] }, { "cell_type": "code", "execution_count": 23, "id": "23107663", "metadata": {}, "outputs": [ { "data": { "image/png": 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enumerate(nn)\n", " mesh = 0:T/n:T\n", " u1 = explicitLinearMultistep( u0, f, T, n, [1], [1]) # Euler\n", " u2 = explicitLinearMultistep( u0, f, T, n, [0 1], [-1/2 3/2])\n", " u3 = explicitLinearMultistep( u0, f, T, n, [0 0 1], [5/12 -16/12 23/12])\n", " u4 = explicitLinearMultistep( u0, f, T, n, [0 0 0 1], [-9/24 37/24 -59/24 55/24])\n", " errs[ℓ,1] = maximum( @. abs( u1 - sol(mesh) ) )\n", " errs[ℓ,2] = maximum( @. abs( u2 - sol(mesh) ) )\n", " errs[ℓ,3] = maximum( @. abs( u3 - sol(mesh) ) )\n", " errs[ℓ,4] = maximum( @. abs( u4 - sol(mesh) ) )\n", "end\n", "\n", "plot(nn, errs[:,1:4];\n", " m=4, label=[\"AB1 = Euler\" \"AB2\" \"AB3\" \"AB4\"], title=\"Convergence of explicit multistep methods\",\n", " xaxis=(:log10, L\"n\"), yaxis=(:log10, \"error\"), ylims=(1e-11,1e5), \n", " legend=:bottomleft)\n", "plot!( nn, (1/2)*errs[end,1]*(nn/nn[end]).^(-1), primary=false, linestyle=:dash )\n", "plot!( nn, (1/2)*errs[end,2]*(nn/nn[end]).^(-2), primary=false, linestyle=:dash )\n", "plot!( nn, (1/2)*errs[end,3]*(nn/nn[end]).^(-3), primary=false, linestyle=:dash )\n", "plot!( nn, (1/2)*errs[end,4]*(nn/nn[end]).^(-4), primary=false, linestyle=:dash )" ] }, { "cell_type": "markdown", "id": "b0b61b02", "metadata": {}, "source": [ "
\n", "\n", "**Exercise 6.** Comment on this error plot with reference to the absoulte stability regions for the particular methods.\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "48ada9d0", "metadata": {}, "source": [ "## C. Implicit Methods\n", "\n", "As we have seen, explicit methods have bounded regions of absolute stability. A justification for using implicit methods over explicit methods is that they can have larger stability regions. In fact, backward's Euler $(\\text{AM1})$ and implicit trapezoid $(\\text{AM2})$ have unbounded stability regions:" ] }, { "cell_type": "markdown", "id": "6d457e34", "metadata": {}, "source": [ "**Exercise 7.** Show that backwards Euler has the region of stability given by \n", "\n", "\\begin{align}\n", " \\mathscr A = \\{ z \\in \\mathbb C : |z - 1| \\geq 1 \\}.\n", "\\end{align}\n", "\n", "(Exterior of a ball of radius 1 centered at $1$)\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "fd03233b", "metadata": {}, "source": [ "It turns out that the stability region of the implicit trapezoid method is the entire left half plane:\n" ] }, { "cell_type": "code", "execution_count": 12, "id": "f400678a", "metadata": {}, "outputs": [ { "data": { "image/png": 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plot([-10, -10, 10, 10], [-10, 10, 10, -10]; \n", " fillrange=-10, title=\"Backward's Euler (AM1)\",\n", " fillcolor=:darkblue, color=:black)\n", "plot!(P1, @. real(ρ1(z)/σ1(z)), @. imag(ρ1(z)/σ1(z));\n", " legend = false, fillrange=0, fillcolor=:white, color=:black, m=2,\n", " xlabel = \"Re(z)\", ylabel = \"Im(z)\", \n", " aspect_ratio = :equal, xlims=(-3,3), ylims=(-3,3))\n", "\n", "ρ1(z) = z - 1 \n", "σ1(z) = (z+1)/2\n", "\n", "P2 = plot([-10, -10, 0, 0], [-10, 10, 10, -10]; \n", " fillrange=-10, title=\"Trapezoid (AM2)\",\n", " fillcolor=:blue, color=:black)\n", "plot!(P2, @. real(ρ1(z)/σ1(z)), @. imag(ρ1(z)/σ1(z));\n", " legend = false, fillrange=0, fillcolor=:white, color=:black, m=2,\n", " aspect_ratio = :equal, xlims=(-3,3), ylims=(-3,3), \n", " xlabel = \"Re(z)\", ylabel = \"Im(z)\")\n", "\n", "plot( P1, P2, layout=(1,2), size=(1000,500))" ] }, { "cell_type": "markdown", "id": "2d315a03", "metadata": {}, "source": [ "
\n", "\n", "The region of stabilty for higher order implicit methods then becomes bounded. Here, we compare them to the (explicit) AB methods (on the same scale):" ] }, { "cell_type": "code", "execution_count": 13, "id": "a3e329c8", "metadata": {}, "outputs": [ { "data": { "image/png": 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ylabel = \"Im(z)\", label=\"AM3\",\n", " aspect_ratio = :equal, xlims=(-8,8), ylims=(-8,8))\n", "\n", "# AM4\n", "ρ1(z) = z^3 - z^2 \n", "σ1(z) = (9z^3+19z^2-5z+1)/24\n", "\n", "plot!( @. real(ρ1(z)/σ1(z)), @. imag(ρ1(z)/σ1(z));\n", " fillrange=0, fillcolor=:lightblue, fillalpha=.5, color=:black, lw=2,\n", " aspect_ratio = :equal, xlims=(-7,1), ylims=(-4,4), \n", " label=\"AM4\")\n", "\n", "xlims!(P, (-7,1) )\n", "ylims!(P, (-4,4) )\n", "\n", "plot( P′, P, layout=(1,2), size=(1000,500))\n" ] }, { "cell_type": "markdown", "id": "197a8449", "metadata": {}, "source": [ "
\n", "\n", "**Exercise 8.** Explain what you would expect to see in the convergence plot above (the plot entitled \"Convergence of explicit multistep methods\") if we also plotted the error curves for the implicit methods $\\text{(AM1)}$-$\\text{(AM4)}$?\n", "\n", ":::{.box}\n", "\n", "***Answer.***\n", "\n", "Your answer here\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "f08091dc", "metadata": {}, "source": [ "## D. Reading \n", "\n", "Make sure you are up to date with the reading: \n", "\n", "* Sections 5.1-5.6 & 5.10 of @Burden " ] } ], "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 }