{ "cells": [ { "cell_type": "markdown", "id": "81b1d2f0", "metadata": {}, "source": [ "---\n", "title: \"Higher Order Methods\"\n", "subtitle: \"Lecture 4\"\n", "date: 2026-02-02\n", "abstract-title: Overview\n", "abstract: | \n", " (1) Error estimates for one-step methods \n", " (2) Higher order Taylor methods \n", " (3) Intro to Runge Kutta methods \n", "format:\n", " html:\n", " other-links:\n", " - text: This notebook\n", " href: L4.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 lecture is mostly based on @Burden section 5.2, 5.3\n", "\n", ":::" ] }, { "cell_type": "code", "execution_count": 1, "id": "db6fe346", "metadata": {}, "outputs": [], "source": [ "#| echo: false\n", "\n", "using Plots, LaTeXStrings, OrdinaryDiffEq" ] }, { "cell_type": "markdown", "id": "5ea01014", "metadata": {}, "source": [ "Recall that we are considering IVPs: seek $u :[0,T] \\to \\mathbb R$ such that\n", "\n", "\\begin{align}\n", " &u(0) = u_0 \\nonumber\\\\\n", " &u'(t) = f\\big( t, u(t) \\big). \\tag{IVP}\n", "\\end{align}\n", "\n", "We will suppose that $f$ is continuous on $[0,T] \\times \\mathbb R$ and Lipschitz in the second argument and so the problem is well-posed and there exists a unique solution to $(\\text{IVP})$. Recall *Euler's* method: we approximate $u$ on the equispaced mesh $\\{t_j = j h\\}_{j=0}^n$ where $h := \\frac{T}{n}$ is the mesh size with the following difference equation\n", "\n", "\\begin{align}\n", " u_{j+1} = u_j + h f( t_j, u_j ).\n", " \\tag{Euler}\n", "\\end{align}\n", "\n", "We saw last week that, if $\\|u''\\|_{L^\\infty} \\leq M$, then we obtain the bound $| u(t_j) - u_j | \\leq \\frac{h M}{2L} \\left( e^{Lt_j} - 1 \\right)$. That is, the error decays with rate $O(h) = O(n^{-1})$ as $n \\to \\infty$. \n", "\n", "::: {#exr-1}\n", "\n", "Prove that Euler's method applied to $(\\text{IVP})$ converges with rate $O(h)$ in each of the following cases (these are the numerical experiments we saw last week):\n", "\n", "* *(i)* $u_0 > 0$, $f(t, u) = \\mu u$, $\\mu > 0$, \n", "* *(ii)* $u_0 = 1$, $f(t,u) = u( 1 - \\frac{u}{20})$, $T = 5$, \n", "* *(iii)* $u_0 = 0$, $f(t, u) = t^2 - u \\sin t$, $T = 20$, \n", "\n", "Write down the corresponding error estimates. Compare your error bounds with the true errors that we observed last week. How pessimistic are the error bounds?\n", "\n", ":::\n", "\n", "A natural question one may ask is \"how can we do better than $O(h)$ convergence?\": \n", "\n", "## General One-Step Methods\n", "\n", "We now consider the following *one-step* method:\n", "\n", "\\begin{align}\n", " u_{j+1} = u_{j} + h \\phi( t_j, u_j; h)\n", " \\tag{1-step}\n", "\\end{align}\n", "\n", "for some choice of $\\phi$." ] }, { "cell_type": "code", "execution_count": 2, "id": "1f847c0c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "oneStep (generic function with 1 method)" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "function oneStep( u0, ϕ, T, n )\n", " h = T/n \n", " t = 0:h:T \n", " u = zeros(n+1)\n", " u[1] = u0\n", " for j = 1:n\n", " u[j+1] = u[j] + h * ϕ( t[j], u[j], h )\n", " end\n", " return u\n", "end " ] }, { "cell_type": "markdown", "id": "ea83165d", "metadata": {}, "source": [ "::: {.callout-note}\n", "\n", "When $\\phi(t, u; h) := f(t, u)$, $(\\text{1-step})$ is simply Euler's method.\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "34ef4799", "metadata": {}, "source": [ "We first define the local truncation error for general one-step methods:\n", "\n", "::: {#def-LTE}\n", "# Local truncation error\n", "\n", "The *local truncation error* of the method $(\\text{1-step})$ is given by \n", "\n", "\\begin{align}\n", " \\tau_{j+1}(h) := \\frac{u(t_{j+1}) - u(t_j)}{h} - \\phi(t_j, u(t_j);h).\n", "\\end{align}\n", "\n", "We say $(\\text{1-step})$ is *consistent* if \n", "\n", "\\begin{align}\n", " \\max_j \\tau_{j+1}(h) \\to 0\n", "\\end{align}\n", "\n", "as $h\\to 0$.\n", "\n", ":::\n", "\n", "It turns out that bounds on the local truncation error lead to a global error bound: \n", "\n", "::: {#thm-global}\n", "\n", "Suppose that \n", "\n", "* *(i)* $|\\tau_{j+1}(h)| \\leq Ch^p$, \n", "* *(ii)* $\\left| \\frac{\\partial \\phi}{\\partial u} \\right| \\leq L$\n", "\n", "for all $(t,u) \\in[0,T] \\times \\mathbb R$. Then, we have\n", "\n", "\\begin{align}\n", " | u(t_j) - u_j | \\leq \\frac{Ch^p}{L} \\left( e^{L t_j} - 1 \\right).\n", "\\end{align}\n", "\n", ":::\n", "\n", "::: {#def-order}\n", "# Order of accuracy\n", "\n", "If $\\tau_{j+1}(h) = O(h^p)$ for some $p$, we say $(\\text{1-step})$ has *order of accuracy* $p$.\n", "\n", ":::\n", "\n", "::: {.callout-note}\n", "\n", "This is a more general version of the theorem we saw last week for Euler's method. In that case we assumed that $\\|u''\\|_{L^\\infty} \\leq M$ and showed $|\\tau_{j+1}(h)| \\leq \\frac{h M}{2}$, giving order of accuracy $1$.\n", "\n", "::: \n", "\n", "
\n", "Proof. The proof is exactly the same as in the case of Euler. \n", "\n", "Define $e_{j} := u(t_j) - u_j$ and note that\n", "\n", "\\begin{align}\n", " e_{j+1} &= u(t_{j+1}) - \\left[ u_j + h \\phi( t_j, u_j; h ) \\right] \\nonumber\\\\\n", " %\n", " &= \\left[ u(t_j) - u_j \\right] \\nonumber\\\\\n", " %\n", " &\\qquad + \\left[ u(t_{j+1}) - u(t_j) - h \\phi( t_j, u(t_j); h ) \\right] \\nonumber\\\\\n", " %\n", " &\\qquad + h \\left[ \\phi( t_j, u(t_j); h ) - \\phi( t_j, u_j; h ) \\right] \\nonumber\\\\\n", " %\n", " &= e_j + h \\tau_{j+1}(h) + h \\left[ \\phi( t_j, u(t_j); h ) - \\phi( t_j, u_j; h ) \\right].\n", "\\end{align}\n", "\n", "Using the fact that $| \\phi( t_j, u(t_j); h ) - \\phi( t_j, u_j; h ) | \\leq L | u(t_j) - u_j | = L|e_j|$ and $|\\tau_{j+_1}(h)| \\leq Ch^p$, we have \n", "\n", "\\begin{align}\n", " |e_{j+1}| \\leq (1 + hL)|e_j| + C h^{p+1}.\n", "\\end{align}\n", "\n", "Applying [A1](A1.html#b-latex) [Section B, Exercise 4 and 3], we conclude\n", "\n", "\\begin{align}\n", " |e_{j+1}| &\\leq (1 + hL)^{j+1} \\left( |e_0| + \\frac{Ch^{p}}{L} \\right) - \\frac{Ch^{p}}{L} \\nonumber\\\\\n", " %\n", " &\\leq \\frac{Ch^{p}}{L} \\left( (1 + hL)^{j+1} - 1 \\right)\\nonumber\\\\\n", " %\n", " &\\leq \\frac{Ch^{p}}{L} \\left( \\big( e^{hL} \\big)^{j+1} - 1 \\right)\n", " %\n", " = \\frac{Ch^{p}}{L} \\left( e^{Lt_{j+1}} - 1 \\right).\n", "\\end{align}\n", "\n", "
" ] }, { "cell_type": "markdown", "id": "9d018bd8", "metadata": {}, "source": [ "**Question:** How to choose $\\phi$ to improve the order of accuracy?\n", "\n", "## Higher order Taylor methods\n", "\n", "\\begin{align}\n", " \\frac{u(t_{j+1}) - u(t_j)}{h}\n", " %\n", " &= u'(t_j) + \\frac{h}{2} u''(t_j) + \\frac{h^2}{6} u'''(t_j) + \\frac{h^3}{24} u^{(4)}(t_j) + \\dots \n", "\\end{align}\n", "\n", "@Burden section 5.3.\n", "\n", "::: {#exr-2}\n", "\n", "Write down the Taylor methods of order $1$ and $2$. \n", "\n", ":::\n", "\n", "::: {#exm-1}\n", "\n", "Let's solve \n", "\n", "\\begin{align}\n", " u'(t) &= u(t) - t^2 + 1 \\nonumber\\\\\n", " u(0) &= 0\n", "\\end{align}\n", "\n", "numerically. You can check that this problem has a unique solution given by $u(t) = (t+1)^2 - e^t$. We use this example to analyse the error made in the Taylor method of order $1$ (Euler's method) and $2$:" ] }, { "cell_type": "code", "execution_count": 3, "id": "9b558f77", "metadata": {}, "outputs": [ { "data": { "image/png": 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TJjr1Qwg5fuxU27Ytr31+wsJNqpapZRnyMcNGjx2n8ynnhJAPhw/vN2DAwYMH79y+3ax587Zt25b5PagUrVu3bty4McMw+H5aJpqm330rBQcHV2xnb6G5M39oHtWSommrunaEEFlqfuq6+L8WL+nciR+Xw93M4G5ncFsSyNcZXEoBF2hF1bWm6lpT9WyoEFvKschUGDs7u/2JqbVnNBNYvvoWZRvqdGLuheYV/WW4devWgb17Ml+khzdr0b17d3zQl4LP5+Mvrkx630qmHoSdO3fesWPHmDFjKIravn17584VP1zP4/HGfDR61fe/+44OMXe34DRsekxi8q74f24/1LWrWrVq7d+xv/97fR+pbgskAlmmPKxu/X0XD1WgKqFQeObMhadPnx4+fNjGxqZ9+/bvcqKoRCLp06dPxU7kgKrO3d393LEz4yd/dmX3FZZwrk4u237bop1lHWBJBVhSfb1fLVmgJnezXkXj4WT2ygvOSkA1dqAi7KkIB0qp4cQBNoUpSAghFLHr5qOM0znAtKc2Xj9xqLeHxEnAO37h8NzpX2/dd9Dd3b3sFwNUFlO/skx2dnbjxo29vb3t7OwOHjx4+vRpf3//Epcs55Vl/v7770lTJ+Tk59EUVS+w7vatu8o5z7NE6enpcXFxYWFh73INaJOlPaiD76elK3qMsIriCHmQzV1I4y6kc+fTuDv/7HFK/tazp1fRZQpS8hwvmu/dskennjf+9de+BTN/aPZfV1dSs5cmU4dPndVH4dUQrixTHjXuyjKWlpaXL18+cOCAQqGYP3++vX3xMwR0NXDgwIEDB+rrEmv29vbvXhKAcVH/DhmH+hFCyBVvvz5ji8/VKkjM9ajVSNee/1q98n91HYu2hDlZKm/Fpaen4w8HTIepByEhxNzcHPv6ACpNWP26dqw0+/YLy7qvTsZQZitS9qftHD9i51+q5o50O1eqgxvlISn7e2RaWppLneJ7QZ0lwufPnyMIwXRUgSAEgEq2d9ueAe8PTDz1QOBuRnI5RULB1pWrI9v63M3i/knhDidz0y5p7MyoSFeqkzvdxpkSv+WDxMPD82Za1vbHz0+mZGg44mwu/Dq0VkJ2ga73+gAwKAQhABTn6Oj4z8Hj9+/fj42NdXZ2DgkJ0Z6HU8eKqmNFfVKHsBxzK5M7kszNv6kZdJxr7kR1dqc7e1Cerw8TB3008r3h73n0D3T7NIiiqYKUvGHLr/vZ18J1wMGkIAgBoGQBAQEBAQElPkVTpL4NVd+G+jyYzlaSw8nsvqfcjGsaBzOqswfV05Nu5EBRhJy5fM61b6Bts1fXtRC7SPynNn764x2WZStwbg/LsgcPHrx945qdg1NUdDSGlaAvCEIAeCeWAtLXm+7rTViOufyC2/2EHXZSk68mvb2ovYeP2I90KrowI+KJnM0fPXrk6+ur01oeP348qGfX+uZcfSv+M4Wm55zv3hs1Zvznk/X6VqCGQhACgH7QFGlkTzWyZ75vSO5kclsfc8/yNA5v3hKLodRqta6dD+3b65s6FsEOFtqH7wVxw9csD4to0rxFi3evHGq4il95BADgbYKsqW9D6cFRzbJvvyjazqrYgqQ8XYeDDx8+tFAXFKYgIYShqLHBDmtXLNNPuVCzIQgBwFC+mfJ17sG0nHsZ2oeqXOXdZfeZyElL79Hpulyf/NmzZ26S4tescLcQJSUm6qtUqMmwaxQADMXFxSXmwIkxEz65ve0WYYhEaL7iy3nmDXusi2OnX1W1c6Xf96M7uFGlXAFcy9XVNTFPWawxIUvm4anbyBKgRAhCADAgT0/Pfdv3chyXnZ1deNZER3cmW8lsfsTOvakZfYaMDKRHBNDO4rd24uPjUyCUXknNDnOy1LaoWW7Rredz1iyshLcA1R6CEAAqQ7GL1loKyIhAekQg/SCbW/2ADd6mauJIjQ9iIl1LvvLh+m07B/bo6vskx0vIKTjqYHLe6IlfNG7cuHKKh+oNxwgBwJj8Lak54czjAfyuHvTEC5raW9SLbrN5b9yX2snJqW2P7vtTX67KzV+TlkU7ObRoE2mMeqEawogQAIxPyicjA+kRgfQ/KdyyWPa7a6r3/OiJdenCK5qOHvfxqfQLATMaUgxNCMl/mhPds+OFf845OzsbtXCoDjAiBABTQRHS1oXaGsnc6MUT0iR0h/qDGM3dLC4jI+PwySNOPbyof89KNPewkEbZz1/yc8VWdOfOnXk/zJo8/tM/161TKBT6ewdQJSEIAcDkuJpTcxsxjwfwG9hSUQc0Xf+4ZeZtWWwZC3/ri1cuVqDzb6Z88WnfbtJTW0KTLtz+fV6T+nVjY2P1UTVUVQhCADBRUj4ZX5eO78eL9hZn57HFntXI1WLR22eavsXBAwduH971Z5RfV3+n5u62Yxu4zW/kNGwAbvRWoyEIAcCkCRkytUuI8FmuuuC1KTQ5F1727tpL197+XrNqRG27oi1+NuaODPvgwYN3LRSqLAQhAJg6Pp+/8IefExfHZse+5NSsIlP+eMsTszSLYe8P07WrtOfPnSVmxRqdxPznz5/rqVioehCEAFAF9OrZ6+i2w7WTPV8seszfrejY+OO8Mcc6HiFXX3A69eNVyycuM69YY1yWzMvLS2+1QlWD0ycAoGrw9/f/e81fhQ9VLFnzgO16WNPMkZrVkPazLPFE/OJGjP1s7MBeaxwsxXxG23L8SYaZo6u7u7tBioaqAEEIAFUSnyYjA+nBvvSi22yzPepeXvTMMMZBVMarQkNDx307q+/X01q7WtrwuasZSo2N89pNWyulZDBRCEIAqMLMeWRaCP1xbXrWdU3wdtVXIczHtenSr+Ldf9DgDp27XLhwIS0trWe9evXq1ausYsFEIQgBoMqzFpKfIpiRgfRn5zXLYtnFTZj2rqXtKbW0tIyKiqq08sDEYbIMAFQT/pbU/mjevEb0yNOarofVT/N0m0cDNRaCEACqla4e9N3evDA7KnSHevpVjUJTwjIymezs2bO7d+9+9OhRpRcIJge7RgGguhHxyPRQZqgfPeE8W3+7elETJtrtvz2lO3fvnDB1kshHwpnTykf5ga7+f63ZYGlZ/BJuUHMgCAGgeqolpXa1Zw4kcp+c1TR3pBY2YawE5NatW+O+/Mz9s9o88av7IyZeet7nvb5H9hw2brVgRNg1CgDVWUd36lYvnpOYBG1V73zC/rx0gXVXl8IUJIRYhzvGpz5OSUmpQOcnY2LaNW0U6u/dIMDn0xEfvnz5Un+FQ+VBEAJANSfikTnhzJZIZspF9sDVB2I3abEFzFzN4+Pjde32zz9WT//4g+9ri3Z3CtjT0a/+85vtmkZkZGToqWqoPAhCAKgRmjpS13vyrGztlFnFb0CoyVLa2dmV+Kq30Wg0c2fOWNHW10366hz+jrXsPvSRLpj7g37KhUqEIASAmkLEI3NHD8s49tr1tWXP86lsEhgYqFNXcXFx/jZiEY8p2tjO0/bk8WN6KBQqFybLAEAN0r17931H9x9d+o95Mxu+VJAfn6u8lL3z7x0UVa5LlRbiOO7NF1AU4TicvFj1YEQIADXLyiUrNi3aEM1rbvfAI4M/rPPS67WD6+vaia+v7/2MfLn6tbMUjz/NaN6qtd4KhcqCESEA1Djh4eHh4eGEkBwVGXlK02S3elNbxtdCh0Ehj8ebOPXLTxbNm93Ew9FcSAg59uTlygfZR//40lBFg8EgCAGg5rLgk41tmXUP2eZ71AsaMwN9dNhJ9uGIUW4eXp9Nm5yXnU0xdGh4xKFTC3WddAOmAEEIADXdUD86zI7qf1xzIJFb3pwRl/tzMSo6Oio62pClQWXAMUIAABJkTZ3rxmMJabxbHZuFCS81C4IQAIAQQqR8sr41Mz6Ibr1PvSOBNXY5UHmwaxQA4D8fBdBhdlSvo5oL6dzshgyt21kVUCVhRAgA8JoQW+pid97FNK7bYXW20tjVgOEhCAEAirMzI4c78uraUI12qe/hkGF1hyAEACgBjyZzwplJwXSLveo9T3HIsDrDMUIAgLcaGUjXsaL6H9c8DCYTgzFyqJ7w7woAUJrmTtT57szah+wnZzWaN/aS5uTkXLhw4cGDB2q12hjVgR4gCAEAyuBuTp3pynuSy3U6qM5RvWpUqVRjJ42r26TesO9Gdh3dK7BBncNHcJv7Kgm7RgEAyibhk11RvE/PalrsUe+LZtzMqdHjPj6Te9l7aj1CEUKIMlsxfOLIvRt216tXrwL9nzt7dtvf6xMTEuqFhY8Y84mDg4Oe3wC8HUaEAADlwlDk12ZM/1p0i72aaym5h2OOOHb0IP+eaCiwFNr0dPv+p9kV6HnSp2O+G/1+w5RLH9vkSi7sim7a6NjRo/osHUqFESEAgA6mhdAeEtL+z3grF4tiT0lrWd3df0fXDo8dPfrozNGVbX20D32tzdt42Lw3avjl2AcCgUAPFUNZMCIEANDNe770j62sX2QUP9lelau0tLLUtbftf69/z9e6aIuNSNDQUXLx4sV3qhLKDUEIAKCzYY09bTQ8eVpB0casU88H9Rmoa1eZGS9tRMVHfrYCOiMj451KhHJDEAIAVMSWNesSl8elnkqRpxfkJWSn/v3YvcBx1PBRuvYTEBR892VescY7WfKAgAA9VQplQBACAFREWGjYnfPXou063dtIqMtOsz+afmTvYR5P54kXH40eszL2ZUquvLDlwKMXxMYZQVhpMFkGAKCCrK2tF38/8/P/cZH7NU98KziucHNzW7Fh06cjPvQS0w5C+nZGgUtA3Q3b/tBrpVAaBCEAwDvxkFAnu/Da7Vfnqsi3oRWJw8ZNmpy7cfvevXuPHj2aFhbm4uKi9yKhFAhCAIB35SwmRzvx2u1XE1LBLGQYJigoyMPDQyqV6rs6KAOOEQIA6IGzmPzTmbf5ETvrOm5VUcUgCAEA9MNBRP7pzPsrjv3hBrKwKkEQAgDojYOIHO/MW/eQnXcTWVhlIAgBAPTJUUQOd2RWxLJL7yILqwZMlgEA0DN3c+pYJ6bVPo2UT4b6Ybxh6hCEAAD65yWljnZkWu/TiHmkjzey0KSZUBDevHnz0KFDaWlpAQEBgwcPFolEhJCsrKzNmzcXLtOkSZPg4GDj1QgAUF5+ltT+Dkz0AbWFgIpypcp+ARiJqXxPycvLi46OTkpKcnFxWb9+fdOmTeVyOSHk+fPnEyZMePSv7OxsY1cKAFBe9W2o7e14759QX37BGbsWeCtTGRGKxeKEhAShUEgIGTNmjIuLy9mzZ9u2bUsIMTc3nzNnjrELBACoiKaO1MoWTLfD6pjOPD9LjAtNkakEIU3T2hQkhHAcp1AoJBKJ9qFCoVi0aJFQKGzbtq2/v7/xagQAqIiuHnRaGOlwUHO2G89RZOxq4A2mEoRFTZ06NSIiIjw8nBDC4/FatmyZlpaWnJz8xRdfrFq1asCAASW+Kikp6dixY8OHD9c+pChq8uTJ7u7uJS4sl8tpmqYofDsrjVwuZxhGo9EYuxCTplKpsInKxHGc9tfJ2IUYzWBP8jib6nSAPdSelfDfuphcLufz3/40EEJ03EoCgYCmyzgIWKlBOHTo0L/++qtYY9OmTU+ePFn4cPHixXv27Dl58qQ2pXx8fPbs2aN9qk2bNpMmTXpbEEokEkdHx4YNG2ofUhRla2v7to3F5/P5fD6CsHQajYZhGPxZlommaWyl0nEcp/2jM3YhxjQ9jKTIuKFnqB2RNO8tn8zYSuWh01Yqz+c8xXEmdAh3+fLlc+fOPXHihKen55vPxsXF+fn5yWQyMzOzN59dsmTJw4cPFy9eXJ4V5efni8ViBGHptF/h8WdZOu2IsMTfSSjEcVxBQYG5ubmxCzEyFUu6HVZ7SKgVzUseHOfm5uKi22XS+1YylVmjhJA1a9b88MMPR48eLZqC2rmjWnv37vX19cUnDgBUUXyabInkXUznfrqFi86YEFM5Rpienj58+PCAgIDRo0drWyZNmtShQ4dZs2YdPnw4MDAwMTHxxo0bGzduNG6dAADvQpEu2qoAACAASURBVMIn+6KZxrs0fhaku+d/QxGNRnP79u2EhIQGDRp4eHi8yypu3bp15fJlc4mkadOmrq6u71xy9WcqQWhlZXXx4sWiLdpx4dSpUyMjI5OSkuzs7CIiIqytrY1UIACAfriIqR3tmc6H1F5Sqr4NRQg5dfrUR2OG0058SsooZhUEuvmv/+1PGxsbXXvOz88fNrBf/tOHTe2EMpb68cucHoPfn/btDAO8iWrFVIKQz+eHhYW92S4Wi1u3bl3p5QAAGFCYHbW0KdPlkOZ8N4bNSBo04j23TwIE1q+O+yRdTevat9uZY6d17Xb8qBFN1Sl9WtfSPhwR7PzFrr83B9bu17/kOYagZULHCAEAao7e3vSoQLrXUc3i5css2tsXpiAhxDrUIU358t69ezp1qFAorpw708ffsbCFoajJoS6/LVmkt6KrKQQhAIBxfNmArmVBbb1wW+xuUewpnpvZw4cPdertxYsXTtLicwmdJGYvX6S/U5U1AIIQAMA4KEJWt2BkQltVtqL4c7kaXadE2NjYpOXLizVmyJQWlpbvUmRNgCAEADAaEY8s/HRo4tHXBm3KbIUsPi8iIkK3rkQin9p1jz/JKNq49Gbq4A9H6KHQas1UJssAANRMAzq23bqn49H5+z3a2wlszGSPc/JiXq5ZtroC17JYtnptn84dYp49bWpvJlOzu5Py/Bu3/HDESEOUXZ0gCAEAjGzrskUTNw3+e9OfDZgnTcPafTxrtK2tbQX6sbOz++f8pYMHD14+d0Yskf4Y3aF+/fp6r7b6QRACABjfz/0bpTuG8TjVV23e6eJZFEV17NixY8eO+iqsJsAxQgAAk/BrM+Z8Ov3HA1x9rbIhCAEATIKET9Y2VX5+QXP9pQndC6EmQBACAJiK2pbckqZMv+OaHJWxS6lJEIQAACZkoA/dxpkaeQp3e648CEIAANOyqAnzIJtbdQ8HCysJghAAwLSYMWRzJPPlZc2tDBwsrAwIQgAAk+NrQc1txAz6RyNTG7uUGgBBCABgiob50/VtqckXcbDQ4BCEAAAmankz5nAyt+sJDhYaFoIQAMBESfhkXStm9GlNcj4OFhoQghAAwHRFOFCf1GE+OKlBEhoOghAAwKRNC6EVGrL0LnaQGgqCEADApNEU+aMVM+Oq5k4mhoUGgSAEADB1taTUd2HM0BiNCsNCA0AQAgBUAaNr084iMus6zqbQPwQhAEDVsKoFb0UseykdO0j1DEEIAFA1OIvJgsbM+zEaOYaFeoUgBACoMgb40LWtqO+vIQn1CUEIAFCVLG3GrLrPXn6BHaR6gyAEAKhKnETkpwjmo5MaJWaQ6gmCEACgihniS9eSUrMxg1RPEIQAAFXPL03pX2PZm7hhoT7wjF0AAADozNWcmt2QGXVac6Yrj6YMsoobN25s/GNN0tPHAXXrfzT6Y2dnZ4OsxgRgRAgAUCV9GECLeWR5rEEOFc6a/s2EQb3qPD49TPzS6tLuzi0a79q5wxArMgUYEQIAVEkUIb82Y5rvUXf3pFzN9TkqvHr16vEtG/5o50tTFCHE31YS6WnXf9JnbSPbSaVSPa7IRGBECABQVflbUh/Xpseff21QeO7cucVLFq9evfrRo0cV63bn5o2DfSy1KaglFfIiXS1iYmLeqVxThREhAEAVNi2EabBDvesJ292TzsnJ6da3e4rmOeNjxqm4mUtm9Yru8fOc+br2mZGeFioSFGu05ZOXL1/qqWrTghEhAEAVJmTI8ubMuHNsrooMG/3hy8AC52G+Di3dHCPdPScF7b62f+2fa3Xt0y8oODajoFjj3VyNv7+/nqo2LQhCAICqraUTFelCfXku/9L1yzaNHP97giL23T1+WbVM1w7fe/+Dvx9lP8n+LwtPJ2UkssLGjRvrpWBTg12jAABV3o8RTO1VLyTWomLtAkthWobO+zNtbW3XbtkxZthQN8ELVzEvNlMudHTbvGcDRRnmRA1jQxACAFR5tkLyZTP7rxfK3F9vV2YrrK1tKtBhg9DQM9du3r17Nzk5+bOAAC8vL32UaaKwaxQAoDoYGyIWujXIuJT2XxNHXuxO/GT4xxXrkKbpunXrRkdHV+8UJBgRAgBUDzRFNv22umufrvI78Xw/M07Fyq/mdG3T5YOhHxi7NFOHIAQAqCba+loPWByTf+9MWN5FC4m07eS2fn5+xi6qCkAQAgBUH/MimKDEJtP6tKhnUz0nthgCjhECAFQftkLyTQPms3O4Q5MOEIQAANXKqNr0SwXZ+QT37S0vBCEAQLXCUGRhE2bSeVaBYWH5IAgBAKqbNs5UsA215C4GheWCIAQAqIbmR9Bzb2hSZcauoypAEAIAVEM+FtT7fvSMq9g9WjYEIQBA9fRNKLP7CXczgzN2IaYOQQgAUD1Z8MlXDehJFzAoLAOCEACg2hoRQCfnk4NJGBSWBkEIAFBt8WjyQzg95aKGRRS+HYIQAKA66+5J2wrJ+jicSvFWCEIAgGpuTiPmy8usTG3sOkwVghAAoJprZE81sqeWxmJQWDIEIQBA9fdDOD3vhiZDYew6TBKCEACg+vO3pHp703Nu4FSKEiAIAQBqhG9DmdX32cR8zB8tDkEIAFAjOInIiED6+2s4UlgcghAAoKaYUp/Z+YSNz8Gg8DU8Yxfwn4ULFyoUr47k1qlTp2vXrtqf4+LiVq9eLZfLBwwY0KhRI+MVCABQtVkJyCd1mBlX2XWtGWPXYkJMaEQ4ffr0hISEzMzMzMzM/Px8bWNiYmJERARFUe7u7lFRUadPnzZukQAAVdrEuvSRZPZuFgaF/zGhESEhZPLkyd7e3kVbli9fHh0dPWvWLEKIXC7/8ccfmzdvbqTqAACqPAmfTAxmvr3CbonEoPAVExoREkLWrFkzd+7cmJiYwpaTJ09GRkZqf27Xrt2pU6eMVBoAQDUxNog+l8ZdSseg8JVKHRFqNBqOK77paZqmaZoQ0qZNG0JIenp6v379Bg0atGDBAkJIamqqvb29dkkHB4fMzEyZTCYSid7sPDEx8dixYx999FFhy+eff+7p6VliJQUFBYQQiqL08K6qL7lczjAMn883diEmTaVSaTQalsVMvNJwHFdQUIC/uDLJZDKGqYyB2sTa9DeXNNtaV8nTCnXaSmZmZtqIKUWlBqG/v39ycnKxxilTpsyYMYMQsmPHDm3LiBEj6tSpM2HCBA8PDz6fr1a/ukCeUqmkafptn8uWlpaOjo7h4eHahzwez8nJSSgUlriwWq0WCoX4sywdx3EIwjLRNK3RaN72mwZaHMdhK5WHUqmsnK00ug5ZfI+7lsNrbF8Ja9MznbZSeT7nKzUI4+Pjy7NYQECAlZXVkydPPDw8XF1dC7MzOTnZ0dGRxyu5ZgsLi7p1644ePbo8q2AYhmEYBGHpmH8ZuxCTph0LYiuVTvulClupTJW2lcQM+TKE/e46e7CDac0UKQ+9byVTOUYok8kK95rGxMTk5uYGBAQQQrp167ZlyxbtZ82mTZu6detmzCoBAKqLD/zp+9nkIo4UljIivHXr1uHDhxMSEpKSknJzc+3s7JycnMLCwrp162Zpaan3Oo4cOTJhwoSQkJCCgoIzZ84sXLjQwcGBEDJs2LA1a9a0bt3a2tr6ypUrmCwDAKAXfJpMrkd/d02zJ6rqDQr1q4T3f+HCha+//losFjdr1qxdu3ZWVlYSiSQrK+vFixe3b9/u06ePvb39nDlzPDw89FhHly5dvL294+PjhULhmjVrnJyctO0SieT8+fMnTpxQKBTr16+XSqV6XCkAQE32oT89+zp7KZ0Lt6/Rx4leC0KO42bPni0QCDZv3mxlZfXm0v379yeExMfHL1iwoGHDhoMHD9ZXHTRNBwcHBwcHv/mUQCCIiorS14oAAEBLyJAv6tGzr7M72hv2wOSjR4+uXLnC5/MjIiKcnZ0Nuq4KeC0If/vttx49egQFBZX+Gh8fnwULFuzevXvfvn2dO3c2ZHkAAGBAIwPpeTfVNzK4+jYGGRSqVKrxo0fcOXequZNYzVKzU3K69B/8zXezDLGuCnstCIcPH17+iZTdunV786RAAACoQswYMrEuPesau9kwF5qZPnWK1aMrG6L8tA8/qe887eDW1R5eH44YYYjVVcxrs0aLpeC+ffuKPrxw4UKxF+P0AwCAqm5MHfrMc+5WhkEGNru2bx1Vz6XwIU1RUxu6rV62xBDrqrASTp84fvz45cuXNRrN+fPni7bz+fxFixapVKrKqg0AAAzOjCGf1aV/uKH/qyMVFBRYCHjM60MmSyE/Py9X7+t6FyXMGr18+fKiRYtyc3Otra2lUmmrVq3CwsJ4PF5oaGhgYODmzZv1OEcGAACMbkwdutYm1cNs2s9Sn/v5RCJRrlJdrFGu1vAEAj2u5d2VMCKcPHlycnLylStXwsLCbt261adPHxsbm86dO8+bN+/AgQMJCQmVXiQAABiQOY+MCqR/vq3nQSFFUY2aNN0fn160cfWd1F79B+p3Re/oredR+vn5BQUFfffdd4SQuLi4mJiY06dPX7hwYfbs2ZVYHgAAVIbxdZmAzapvGjDOYn12O3/p8t6doi+kP23tJFax7P7kAsbN98//TdPnOt5ZaRcU6Nevn/YHX19fX1/fojd2AACA6sRWSAb50ovvaH4I/2/6qFKpfP78uYuLS4Wv7WllZXX0zPkDBw6c/eeYUCT6bHIHE7yn7GtBmJyc7OrqWviwxNPbS1keAACqri/q0Q22q6fUZ6wEJCUlZfinI+48uCu0FhWk5fXo3H3e93PF4oqMFimK6tSpU6dOnfResL68dozw8uXLK1euLM/LlErlN998k56eXvaiAABQFbibU53c6eWxrEwma9WhTUrtLK+pwc6jfX2+Djmec7Z7/57GLtBQXgvC7t27+/j4DBo0aOfOnUqlssQX5OXlrVixYtiwYQMHDgwJCamUIgEAoDJMDaEX3das/nMDEyyyrGP7qpUi9m1cH2c9uXXrllGrM5TixwgjIyObNGmyaNGiuXPnCgSCgIAAW1tbCwuLjIyMzMzMGzduEEJGjhy5bt063FoMAKCaqW1FhdvTG/efM/ORFHuKV8vsxo0bZR4yq4pKmCwjFounTp06derU+Pj42NjYZ8+eZWZm1q1b18nJacaMGTgoCABQjU2pT3fLF7kqNMXaKSURiURGKcnQSps16uPj4+PjU2mlAACA0TVzpFzCu2cfPWZdz76wkdOw+TezWi5uacTCDEeHO9Rv2LBBu2sUAACqsRmDI+VUnWcb4uXpBRzL5T3JebokdtzIsfb29mW/uAoqLQjXrVs3atSo9evXp6SkEEIGDx6cl5e3Y8eOyqoNAACMoIcnLR29aWjvb8yPcGnz41yuWW5e+vfn4ycZuy5DKW3XaPPmzePi4lauXDl8+HBvb+/mzZv7+/sXFBT07FltJ9ECAABNkXFB9JnnfY7v62/sWipDaUFYq1atmTNnEkJkMtn58+dPnDixZcuWXbt2VVZtAABgHB8F0N9fUz3KpWtJq//t9sp1jFAkErVp02bGjBnbt28/c+aMoWsCAADjMueRYf700rv6vzeTCSotCG/fvj1q1KiVK1empqZqW9zc3LKzsyulMAAAMKZxQfTaB2x2yddWqVZK2zW6c+fOoKCg3bt3jx8/vmHDhg0bNuQ4zsXFpZSXAABA9eBqTkW70b/fZycG63B+QVVU2tsLCAjo0aPH3r17k5OTR44cKRKJAgICPvvss0orDgAAjGhiML34Dquu7vtHSxsR9u3bd9OmTYcPHx4+fPiQIUMqrSYAADAFYXaUp4RsT2D71arOg8LSgpAQ0r9/jZg7CwAAJRpfl/75VjUPwur83gAA4B1196RTCsjlF5yxCzEgBCEAALwVQ5FRgfSyan0eBYIQAABKMzKQ3vmETZMZuw6DQRACAEBprIWkpyf9+4NqOyhEEAIAQBnGBtFL77KqahqFCEIAAChDiC3lLSW7n1TPJEQQAgBA2cbWoZdU0ykzCEIAAChbTy86LofczaqG51EgCAEAoGx8mnzkT/1aHQeFCEIAACiXUbXpv+LZXJWx69A3BCEAAJSLi5hq6URvjK9ug0IEIQAAlNfHdarh3XoRhAAAUF7tXSmZhlxIq1ZTZhCEAABQXhQhIwLpX2Or1aAQQQgAADoY5k/vfsq+VBi7Dv1BEAIAgA5shaSLO/3nw+ozKEQQAgCAbkYE0ivusdXmOCGCEAAAdNPCiaIJOfe8mkQhghAAAHT2YQC96n412TuKIAQAAJ194E/vTGCzlMauQx8QhAAAoDNbIYlyo/+uFleZQRACAEBFjAigl1eLEwoRhAAAUBGRrlSBmlx5UeWnzCAIAQCgIihChvnTq+5V+UEhghAAACroA39qy2O2QG3sOt4NghAAACrIRUxFOFA7Eqr2oBBBCAAAFTfMn17zAEEIAAA1VXdP+mYG9yi3Ck+ZQRACAEDFCWgywIf+82ElBaFMJouLi1Mo9HnzCwQhAAC8k2H+9B8PWUNfhDs1NXVgz24t69X+fsSgxnX8PnpvYEZGhl565umlFwAAqLEa2FJWAhKTyrVxpgy0CpVK1T0qcpyvuFXnQG3L3rj7vTpFHT97kabfdUSHESEAALyrD/zoNYa8BvfuXbsiLKlWHraFLV18HTxJQUxMzLt3jiAEAIB3NdiX3vOUzVEZqv+bVy6F2giLNYZa8W5cu/bunSMIAQDgXdmZkbYu9NbHhhoUiqXSPKWmWGOempNIpe/eOYIQAAD0YIgf9edDQwVhh85ddyflFZ2Oo+G4fYl57dq3f/fOEYQAAKAHndzpO5lcgmFOKKxfv36DyE5jT8Tff5mn0LC30nJGHovr9t77Xl5e7965Cc0anTt3btGH4eHhbdu2zcjIWLVqVWFjZGRkw4YNK700AAAog4AmfWvRf8Vz00IMMnd07sLFx44e/e2XhQk3H/v5+U9f+VOTpk310rMJBWFmZmbhzwsXLly+fDkhJD09febMmWPHjtW2y+Vy4xQHAABlGeJLvx+jmRZiqH2Nke3aRbZrl5ubK9XHocFCJhSEc+bM0f5w5syZpUuX9unTR/vQ3Ny88CkAADBZjR0oipDLL7iGdoY6odAQTCgIC/3+++8DBw6USCTah3K5fObMmWZmZlFRUSEhIcatDQAASjHIl/7zIdvQjjF2ITqo1CBUKBRqdfH7VjEMY2ZmVvgwLy9v69atR44c0T4UCARdu3bl8XiJiYktW7ZcuHDhhx9+WGLnT5482b9/f3JysvYhn8//5ptv3nYctaCggBBCUVXpO0vlk8vlDMPw+XxjF2LSVCqVRqNh2ap99X1D4ziuoKAAf3FlkslkDFOVIuRN/dxIy0O8mcEKvsHmYuq0lczMzMq89EylBuGnn366efPmYo0RERGHDx8ufLhp0yZXV9eIiAjtQ29v7w0bNmh/btq06YQJE94WhDY2Nn5+fv379y9scXNzKxqxRWk0GjMzM/xZlglBWCaGYbS/TsYuxKRxHMeyLLZSmVQqVVXfSv5mJMCKPfnSrLO7oVah01YqzwXYKjUIV61aVXQKaIl+//33ESNGlPhUeHh4WlqaXC4vcRNIpVI/P79+/fqVpxKapmmaRhCWjv6XsQsxaTRNcxyHrVQ67SbCVipT9dhKQ3zJ+niuq6ehhrZ630qmtcXv379/5cqV9957r7AlJyen8OfNmzcHBARU9a9LAADVW79a9OEkNltp7DrKzbQmy/z2229du3Z1cHAobPnxxx+3bdsWEBCQlJSUnJy8adMmI5YHAABlshKQti709gR2mL9pjbXexrSCcMKECWKxuGjLt99+26NHj6SkJHt7+5CQkGLPAgCACRroQ6289yoIL126NPaLcc/SUilC/Hz8ls7/xd/f39gFvsa0gtDFxaVYC4/HCwsLCwsLM0o9AABQAV086FGnNakycvP04eGfj3Qa6uPlFEwIeRmfFdm9/aFtB+rUqWPsGv9TNcatAABQhZgxpIsHveURO+F/E11HBYiczLXtUh8r+0GeE6dNMm55xSAIAQBA/wb60H/dl+cp8wWWr91HUFrL6v6D+8aqqkQIQgAA0L92LtTjPKLWlHAzCoPcn+IdIAgBAED/eDTp7SskfEvFS1nR9pyHGcFBdY1VVYkQhAAAYBADa9Fm/RYmr3iQ//TVGeE5sS9fbkxc8MPPxi2sGAQhAAAYRDMnivVtsWTVXsvTvKezbyf+cNc73unMkdO+vr7GLu01pnX6BAAAVBsUIf28qUt0nWP7jhq7ltJgRAgAAIYywIfeGM+Z2uyYYhCEAABgKA1sKT5NLqebdBQiCAEAwID61qK2PjbpG3YiCAEAwID6etObH5v03lEEIQAAGFA9G0rEmPTeUQQhAAAYVh9vaosJ7x1FEAIAgGH19aa3mPDeUQQhAAAYVrBp7x1FEAIAgMGZ8t5RBCEAABicKe8dRRACAIDBmfLeUQQhAABUBpPdO4ogBACAytDXm95qkntHEYQAAFAZgm0oAU2uvTC5KEQQAgBAJenhRe14YnJ7RxGEAABQSXp60tsfY0QIAAA1VSMHKldF7mWZVhYiCAEAoJJQhPTworYnIAgBAKCm6ulF70gwrcOECEIAAKg8LZ2op/nc0zwTGhQiCAEAoPIwFOniTu8wpb2jCEIAAKhUPb1okzqJAkEIAACVqr0rdeMllyozdh3/QhACAEClEjKkgzu9x2QGhQhCAACobD09TegSMwhCAACobB3d6TOpXJ7K2HUQQhCEAABQ+aR80syJOpRkEoNCBCEAABhBNw9691OTOIkCQQgAAEbQzZPan8iqTWBMiCAEAAAjcBFTXhLqzHPjDwoRhAAAYBzdPOk9T40/JEQQAgCAcXTzpHY+wYgQAABqqvo2FMuRu8a+PSGCEAAAjKarB7Xb2INCBCEAABhNVw96t7EvMYMgBAAAo2nlTN3PNvIFuBGEAABgNHyaRLvR+4w6dxRBCAAAxtTNgzLuJWYQhAAAYEzRbvQ/KaxMbbQCeEZbswnIzs52dHRUKBTGLgRMC5/PT0pKcnBwMHYhADWCtZCE2FIxqVwHN8ooBdToIJTL5VZWVqmpqcYuBEyLt7d3QUGBsasAqEE6u9P7nrId3BijrB27RgEAwMg6e1B7jHeYEEEIAABGVteaoiijXWIGQQgAAMbXyZ3aZ6RBIYIQAACMr7M7vS/ROGcTIggBAMD42jhT115wWUojrBpBCAAAxifikRZO1OEkIwwKEYQAAGASOnvQ+xKNcJgQQVidqVSq//3vfyqVytiFAACUrYsHdSCRZSs9ChGE1ZlarZ47d65abbwrFwEAlJu7OeUooi6lV3YSIghNSE5OzmeffRYaGhoZGbl3715CSEpKSpcuXRITEwkhCoViwIABp0+fJoSsWbOmTZs2QUFB3bp1u3btWmEPf/zxR2RkZFBQUN++fbOyssaNG0cI6dKlS/v27e/cuWOktwUAUF6dPajKnztaoy+xVqIsJeEM/3WERxMp/7UWjuO6d+8eHh6+f//+p0+f9unTx9nZOSwsrHnz5n379j116tTEiRPVanXz5s0JIba2titWrLC1tT148GD37t3v378vEomWL1++YMGCtWvX+vj4XL16leO4KVOm/Pbbb0uWLDEzM3NxcTH4uwIAeDed3OlJ5zUzwyp1pQjC11x/ybXdX0k7Ene157Vw+u8Kszdu3IiNjd22bRtFUX5+foMGDdq8eXNYWNjkyZNjYmI6duz46NGjq1evahfu1q1bbGzs2bNnBQIBy7KxsbGhoaFLliyZN29e48aNCSHR0dGEEDMzM0KIt7e3SCSqnDcFAPAumjpQ8blcmow4VOKHlhGCMDs7+/z583fu3LG3tx8yZEhhu1KpXLp06fXr1wMCAsaPH29ubq5tv3Xr1sqVK+Vy+cCBA9u2bWvQ2kJsqYwh/LKXM4DHjx/n5eVFRUUVtnTo0IEQQtP0559/3q5duyVLllhZWRFCOI4bMmTIgwcPWrRowefz1Wr1ixcvCCFPnjwJCAgwSvEAAHrBo0lrZ/pIMjvYt/KO3BkhCNetW7d+/Xo+n8+ybNEgHDVq1KNHjz755JM///zz9OnT+/fvJ4Q8fvy4RYsWU6ZMsbe379Onz9atWw2dhcbi5uYmlUovXLjAMK9dfz0vL2/MmDHjx4+fNWtWr169XFxcEhISDhw48Pz5cx6Px7LsmjVrtEu6uro+evQoMDCw8LU0TRNCuErY1QsAoCfRbtShJG6wb+Wt0QiTZcaOHXvhwoWPPvqoaGNKSsrff/+9cePGfv36bdq06eTJk7du3SKELFu2rEePHlOnTh0+fPi0adN++umnyi+4coSGhnp6en7xxRfZ2dkKheL8+fPaHaFjxoxp2bLlwoULR48ePWjQII1GIxaLZTLZ3bt3lUrlzJkzMzMztT2MHj162rRpDx480Gg0ly5dysnJEQqFjo6OBw8efPTokVwuN+r7AwAol07u1MGkSj2JwlRmjV66dMnHx8fZ2ZkQIhaLGzVqdO7cOULIuXPnWrVqpV2mdevWZ8+eNWaVhsQwzJ49e/Lz88PDw319fWfOnMlxXExMTHp6+qJFiwghX331lYuLy9atWx0dHRcuXNirV6/atWvzeLwBAwZYWFgQQsaPH//+++/379/f3d39yy+/1J41sWbNmrVr1w4YMOD+/ftGfocAAOXgbk7ZmVHXX1ZeEhpk16hCoXjzvqYURWkPcZUoNTXV1ta28KG9vf2zZ8+07XZ2doWN2dnZBQUFYrH4zR4SEhL279+fnJxc2DJjxoxatWqVuDpteTKZrLxvqVLY29uvWLGiWGPh9wCGYf766y/tzyNHjhw5cmSxJWmanjBhwoQJE4o2duzYsWPHjoapt9riOE4mk5Xz3rwqlUqj0bCscS4WXFVwHFdQUEBRxrn/eBUik8mKHRypmdo5MXseawLFmhKf2ljSjgAAG4ZJREFU1WkrmZmZaQ8SlcIgQbhu3bovv/yy+Jp4vJSUlLe9xMzMrOgFUBQKhXbGo1AoVCqVhY00TQsEghJ7sLe39/Pz69+/v/ahQCDw8vLSdvImjUZjZmYmFArL/Z6gBqEoSigUvu2XpxiGYbS/ToauqkrjOI5lWWylMqlUKmwlQkgnT/LDDfbrsJKnLuq0lcpMQWKgIBwxYsSIESN0eombm5v2tHGtp0+f9u3bt1h7YmKis7Mzj1dyzebm5n5+fv369SvP6uh/6VTkmzQazd27d58/f25vb1+3bl18las2yv/rQdM0x3Hv/rtUvWk3EbZSmbCVtFq7kP7HNblq2rKkgY/et5KpbPHmzZsrlcrjx48TQu7cuXP//n3tDr1evXpt3LhRe7hrw4YNvXr1MnKh/9JoNPPmznV2sG/UMGz4oH5NIho52tl+/93MyrmwZ1JS0o0bNyr88rt376anp79LASzLKhSKN9vVanXRdplMdvHixXdZEQDUQGYMaepIHUuppCMORgjCEydO+Pj4TJky5dq1az4+PqNHjyaECIXC+fPn9+/fv2fPnpGRkbNmzbKxsSGEDB06lMfjNWrUKDIyMiYmZvLkyZVf8JvUanXvHt1XzJ/zY3Pv2FFtTg9qFDuq9aLWvn8uXdS5Q3ThvlydbN++3fF1qampb1t43759s2bNqljxWVlZ3bt35/MreLrknTt3WrVqJZFIbGxswsPDCy/wxnHc559/bmtr6+DgMHDgQO0kVTMzs3Hjxt2+fbti6wKAGivajT6UVEnzZYxwHmGjRo2OHDlS+LDwxPkhQ4a0adPmzp078+fPL5zkIhKJYmJiLl26JJPJmjRpYiJ7z5ctXXrj/JndvUOtzV7FCUNRrTztdjpZ9dhxbf5PP02dNk3XPmUymYODw4kTJwpbrK2t9VVwUcuWLevZs2cpE5dKp1Kpxo4d26VLF4FAMG3atH79+j18+JAQsnv37m3btsXFxUml0nbt2i1evHjy5MkURY0ZM2b27NmF03wAAMqjgxs1/1b1HRGKxeJaRTg6OhY+5ebmFh0dXWyqJ8MwjRs3btOmjYmkICFkycKfP2/oUZiChaRC3v8aeS1ZtKBi57AzDGNbBE3TarW6ffv2hTsbZ82atW/fvmKvio2N7dGjh6+vb/v27S9fvqxtfP/99zdu3NiyZcvg4OBiy69Zs0Z7/JUQsmjRol9//XXw4MFeXl49e/YsZQxaKCQkpE+fPtqJWB988EF8fLx28Ld27doPP/zQ3t7ezMxs/Pjxa9eu1S7fs2fPXbt2ZWdnV2CDAECNFWhFCWhyN6syBoW41qjOXrx4EZfwtFV0yZc9aOVp+3zvtYSEBG9vb117zszMLMwPMzOz/v37syx79OhRjebVHOJbt24VnkyilZGRERkZuWTJkk6dOsXExHTr1u3OnTvW1tZnz569d+/emjVrio38EhMTExMTQ0JCtA/v3bu3c+fOTZs2/fLLL2PGjJkxY8avv/5KCBkzZkzhefqFevbsWWwu0q5duyIiIrRfUOLi4gYPHqxtr127dnx8PMdxFEVJpVI/P7+zZ8/iLA4A0EmUG3UwkatjZfCzbhCEOsvKyqIoykJY8qYTMLRYKMjKyqpAz3l5eSdPntT+LJFICk8FKcXGjRvDwsLatm0rl8sjIiKCg4OPHz/eu3dvQsjnn39ep06dYssnJCQ4OjoWPUDYr1+/li1bEkI+/PDDr776StvYtWvXN0+yLHrxNkLIyZMn586d+88//2gfZmdnF+7llkgkCoVCJpNpz/h0dXVNSEgozxYAACgU5UqtuMdODDb4nksEoc7s7e0JIekFSkfzEk5DzFWo8xXKovt7y8/d3f3333/X6SVPnjy5efNm0YFa4SkcHh4eby5PUVSx3baFpYrF4vz8fO3P2ptaFHtt0RNXLly40KdPn82bN9evX1/bor3cgfbnzMxMiURSeN0DXOwUACqgjQv9QYxGoSFCA5+YhiDUmaWlZVi94H0PUz8M8Xzz2X1xqf61vPR18z8ejycQCPLz87Wh8uYVCTw8PIKCgrQXKC+mxKt4eHh4pKWlqdXqt52OqbV+/fqMjIxijf369fP39yeEXL9+vUePHr///nu7du0Kn61Tp87Vq1e1o9hr164VHYw+e/bM07OEbQUAUAorAaltTZ1N49o4G3bvKIKwIr6cPmPYe4OaudsG2EqKtj/OKph7MWHhsuKXSSunvLy8wj2NhJCGDRtKpdLQ0NBly5Z98skn+/btu3z5cuFxOK0BAwZ89913y5YtGzRokEKhiImJadasmaur69tW4eHh4ebmduPGjbCw0m58WXhHizfdu3cvMjJy8ODBIpHo6NGjhJBmzZqJRKKRI0f26NGje/fuNjY28+bNKzzRJS8v78GDB02bNi3nRgAAKNTelTqSxLZxNuyQEEFYET169Lg2YVKP+T+ODvHo6OPgKBGmFygPxactu/Z0xMjRxbKqnJydnb29vWfPnl3Y8ssvvwQEBKxatWrcuHEbNmzo3bv3//73Pzc3N0KIm5tbvXr1CCG2trZnzpz55ptvfvnlF4ZhGjVqpL1NVdOmTS0tLUtc0QcffLBlyxZtEAYGBjo4OGjbraystDf1LV1ycnJoaGhsbGxsbKy2Zf369SKRqHnz5j/99NOYMWNUKtWQIUOGDRumfXbnzp1du3at8NkaAFCTtXelJ53XzA438Gq46mLx4sVjx44t58J5eXksy6ampjo6OlZ4jYcOHYps1ULA5xFC+DymdfOme/furXBvlSYjIyMgICAnJ6dyVte4ceObN29Wzrr0xcvL6/Hjx+VcWKlUymQyQ5ZTHbAsm5eXZ+wqqoBK+8OsKpQaznKtMv31vzC9byWMCCsuKioqKipKrVbn5uZKpdLSj7qZDmtr623btikUCqlUauh1yeXyBQsWvHkuIwBAefBp0tyR+ucZ29ebJoSciDmxaPniR48f/b+9e4+HMv37AH7NjGWIjGMG0zpFUosolu2RGiUhamtrd0vksBX7SrVpkdJx205bq7Wxbdl42tTudnxIm0OWdBCrLZRDGCKmwZhhmmGeP+7fa15+TjnMdt9jvu+/Zm6X6/42vPq4T9/LfIr5V2FbZ8+eLZW9EKXXqIx68+bNgwcPcnNz7927J0Mr31pZWfV5HvFfQqVSh3O6FQAABuNmQL5VL0YIhUdsXhcdXDudrRakXznl5YrQVXsPjrLTZB8QhKMkFAp3xOzQ1Nac784M+HLdgsULNbW1IrZH4BWHDx8+rK6ufgc7amxsxHqjj45AIOByuUMMaGlpycvLG/X8CCGxWPz69Wtxv2c2RCIRm82WdCdACOXk5PTvGwAAIBQ3Q1IGS1xSUnI5+6phyFRVI3UKVUHNVGNy2LTE//2ppqZm7LuAIBwNoVC40MP95C/xpl9azzo532rfh/Zx88w32/506Ywrc97osjA1NVWtHzqdPsxvP3r0aFpa2ij2O1JFRUWS5+5H5MmTJw4ODjQazdjYeNq0afn5+QMOi4iIGMtvtr+//8SJEy0tLTU0NPbu3SvZfvPmTQaDYWNjY2xsLAnasrKyXbt2jXpfAIB3YBqN1C1GSX9cU7ZTR70eoyCRScoz1Xt3rh41CMLR+O6774rKiy2jZtEstf7zgyEhdQtNy8hZzxqeH/jmwCjm/Pjjj5ubm5ubm5OTk2k0GvZ6nDVkiYmJ4XK5zc3Nn3322YB9c2prazMyMobTUmcwjo6ONTU1TU1N+fn5hw4dun37NkJIIBCsWbMmPj6+rq5u9+7dfn5+WLsAPz+/CxcuvHr1atS7AwC8A/P1SU8a2ygq/e7DoJLbuO1jnx+CcDS+izuu72uioNK36TZFiaK/3PT7k3H927K8FZlMplKpVCoV63+GvY6JibGwsDAwMGAymdhiRvfu3fP19ZV812+//RYWFtZ7HrFYfOLEienTpxsbGwcFBbW3tyOEamtrmUxmfHy8iYlJSEhI7/EikSg2Ntbc3HzKlClbt27FDmcfPny4atWqPXv2MBiM2NjYzs7O4ODgyZMnOzo69l4H8dmzZ97e3gwGY9asWenp6dhGf3//U6dOzZkzR0tLC1tIEmNlZbV48WIFBQUSibRy5cr6+npJIxuJ5ORkLy8vrDnO06dPlyxZcvz48ffff9/c3Dw1NXU4H2NISAi2gNe0adNmzJhRXl6OEMrIyFBVVfXx8UEIrV69uq2t7e7du9iHvGDBgvPnzw9nZgAAXpgGJI7OTOGLvn0fxbVCW2ubsc8PQThir169aqitp00f+GYTjWnara850jqSYzKZhYWFdXV1kkMoe3v7oqKie/fuYQOOHTvW526UixcvHjt27PLly48fP+ZwOBs3bkQIdXV1ZWdnl5aWFhcXHzx4sPf477///sqVKzk5OQUFBQ8ePIiNjUUIcbncS5cukUiksrKyTZs27dy5k8ViPX36NDU1VdIWvKOjw83NbdWqVS9evEhISPD398cuUpaXl8fFxSUmJlZVVUn6vfWRmprq6Ogo6U0qkZ2dLbkNjMfjpaWltbe3P3v27IcffggKCuro6EAIpaSkLOln7dq1faaqrKwsKSlxcXFBCFVVVVlYWGDbKRSKmZlZVVUV9nb27Nk5OTnD+2kAAPDhZkB+buyFKkWt/7RINr4ufKXKUZo7d+7Y55eNO/4JhcvlkkgkBeWBPzqSAlmRqiitVYdcXV2zsrLq6upEIlFFRQWbzdbS0goMDExMTHRwcCgtLS0rK8NabEukpKR8+eWXZmZmCKEDBw5YWVlhbWLEYvE333wj6f/Ze/zXX3+NXYzctWuXn5/fgQMHEEI0Gi0yMpJMJiOEzp8/f/HiRVVVVVVV1Q0bNmCHUDdu3NDV1XVwcKipqVFXV//oo4/S09PXr1+PEAoNDe3Tobu3zMzMI0eO9O6hI/HixYvebXFUVFSioqLIZDKTydTQ0KisrLS2tnZwcOh/y6uS0n/1fW1ra1u+fPmWLVusrKywt73/4aqqqpKu6NAQHADim6SMDNWVDvzyZ1xM4NMb/1B1lTsbeQ42s3+8Fo/9HzVGEIQjpquri0hIwBEoaQ6wPqKIJxTwBcO/yWUIPB7P2dl56tSpM2fOpNFoCgoKHA4HC8KpU6cePnz41KlTfn5+fZZpbGpqYjAY2GsGgyEUCltaWhBCGhoa/VMQIdTY2Nh7fGNjI3a/pb6+vuQ3rKmpSZJPWGsbhFBdXR2Lxep9olVZWRl7MUSPt7y8vJUrV166dGnAhwspFErvuzqxRRmx1yoqKnw+HyEkEAiw87299f4QOjo6Fi9e7OTkFBMTg23R1dXtvR4Ih8OR9NPp6ekZ7LAVAEAcbgakIpFe2h//x+VyS0tLp0+fPuB/aKMDQThiampqNjNtWgoaDDxM+n+1+V7DFMspenp6Y99Rfn6+goLCr7/+ihBqaWn54osvsO16enoLFy78+eefz507l5ub2+e7DA0NKysrsdcVFRVUKlVXV7e9vX3AHtyS8dj51crKSkNDQ2xk7/EGBgZVVVVYXkpOKhoZGdHp9AFv2RpsXwUFBT4+PklJSYOdzTAyMurfWLyPJ0+e/PHHH3020mg0Ly8vhBCfz/f29rawsDhx4oSkjBkzZkRHR2Otxvl8/tOnTyUxXF9fb2RkNPQeAQC4m69PPlzSHW1LVlNTs7S0lGIKIgjC0dkbu3fZJx+rT9NSNfqvfp58Frf+t8qziYN2rB6RSZMmVVdXFxcXa2hobNu2rXfnmvXr1y9atMjR0bH/ioOBgYEBAQHOzs76+vrh4eEBAQFDnzoICgras2ePlZWVoqJiZGRkUFBQ/zEBAQFRUVFJSUlsNvvHH3/EFqLy9PSMjo6OiorCDgrz8vJsbW2HOCNaWlrq7u6+bt06RUVFrFu3k5NTn99mV1fXgoKCgICAIQpesWJFn/WBe/Px8WGz2du3b8cedjQ2NjY1NXVycjIwMIiMjAwODj5y5Mjs2bMln1tBQYFUrjEAAP5V/6NHWpkp7hShQa5KjQkE4Wh4eHjsiIyO3R1LX2CkaT9JUYMqbBO8Lmx6efPF1s1bly9fPpbJsQM+hNAHH3wQGxsbGBiorKy8efNm7BIdNsbFxYVGowUHB0u+y97e3sTEBCHk7u5+/Pjx2NjYtra2RYsWRUZGIoTU1NS8vb0H3N26detEItGmTZuEQuGnn34aHh6OENLV1XVzc5OMiYyMFAqFK1asMDQ0PHjw4KNHjxBCVCo1Nzd39+7dy5YtI5PJNjY22AK/8+bNG/DMMIfDWbBgQV1dXUJCAral/8mNzz//3N7eXigUvvfee5qamh4eHpIvLVy4UEtL662fno6ODo1G++mnn7C3S5cuNTU1JZFIV65c+eqrr3x9fa2trbGDbIQQn8/PzMyMi4t767QAAHypvoestUh/NYndDP6FJZmk27oUR+++6fadO3fcF7tTVagIIaoKlenulpmZOerZRiQ3N5dOp3d1db2b3b1LGzduPH369LvZ1/HjxyMiIvpvh6bbUgdNt4cJmm4PIeah6Ov7IjE03SaUOXPmpM1JQwjx+XzpnrAe2tq1a69fv37s2LE+t0qOD/v27Xs3veIQQs7OzpLHKgAABDdPn7ztfvfbx40cBKEUvMsURAjt2bPn6NGj2GPj44+6urqNjRSekB2OoVcnBgAQyoeTSGWt4tY3SOr3ecMD9bKHwWCM1xQEAIDBKJKRoy7pzssR9+16KwhCAAAAssFVn5z1su/CMmMHQQgAAEA2uNJJmQ0QhAAAAOSVnTaptkPcIpDyExTyfrOMUCgsLCzEuwpALAKBAO8SAAADUCCjj/RIf70iGQ+86sFop5XmZLJGVVXVwsKiz7JEQEIsFqPB+6WNY5MnT6bRaHhXAQAYgCudnNtMWS3VOeU6CCdMmDDYOukAIdTV1UWhULD1EQEAgAjm6ZNOlUr5oh5cIwQAACAzrLVIwVOk/Fg9BCEAAACZQUIo2Ewk3TkhCAEAAMg1CEIAAAByTU6DsKCggMPh4F0F0f3zzz+1tbV4V0F0NTU1T548wbsKonv9+nVBQQHeVRCdQCDIysrCuwoZkJ6eLt0J5TQIDx8+DPeLvtXZs2evXbuGdxVEd/Xq1bNnz+JdBdHl5eUdPXoU7yqIrqKiYvv27XhXQXRdXV1r1qyR7pxyGoRgmLBHCcEQ4CMCQNZBEAIAAJBrEIQAAADkGmncnNjZsmXLpUuXzM3NhzO4uLiYwWBoaWn921XJtPLychUVFQaDgXchhFZbW9vZ2Qkr3Q+tpaWFxWK9syWXZRSPxyspKfnwww/xLoTQenp6cnJyXF1dhzne19d3w4YNQ48ZP0FYWVl5//59HR2d4Qyur6/X1tZWUlL6t6uSaS0tLUpKSmpqangXQmhcLlcgEGhrS7UH8LgjEAjYbLa+vj7ehRBaT09PbW2tkZER3oUQXXV1tbGx8TAHGxsbm5qaDj1m/AQhAAAAMApwjRAAAIBcgyAEAAAg1yAIAQAAyDUIQgAAAHJNHhfmZbFYmZmZ2trabm5usOrsYFpbWx8/fqyjozN16lS8ayGukpKSkpISNTU1V1fXiRMn4l0OEXV1dT169KiyspJKpTo7O8ONo0NraWkpLi62s7PT0NDAuxYiysrK6u7+z2KEdDrdyspKKtPK3V2jeXl53t7ePj4+paWlioqKf/75p4KCPP41MLSwsLCEhARlZeVVq1bFx8fjXQ5BhYWFpaenOzg4NDc3P3r0KDMzc8aMGXgXRThJSUkJCQnm5ubt7e23bt1KTk729vbGuyji8vT0vHnz5q1bt+bOnYt3LUQ0YcIEOzs77Mk3Nze3bdu2SWVauQtCJpPp7u6+devWN2/eWFtb79+/39fXF++iCIfFYuno6ERERAgEAgjCwVRVVRkZGZHJZIRQYGBgZ2dnSkoK3kUR2qFDh65evZqbm4t3IQT1yy+/ZGdn37hx48KFCxCEA5owYUJ5ebmhoaF0p5Wva4R8Pj8zM3PZsmUIIUVFRW9v7+vXr+NdFBEZGhpCt4G3MjExwVIQIUSn0wUCAb71EB+Px4POA4NpbGzct2/ft99+i3chRHf//v3bt2+z2WwpzilfZwVfvnwpFosNDAywtwYGBn///Te+JYFxoLGxMTEx8cyZM3gXQlAsFsvf37+1tZVCofz+++94l0NQoaGhu3fvhj8UhqatrX369OmOjo6ioqKTJ0+uXr1aKtPK1xFhd3c3iUSS/BVPoVBEIhG+JQFZ197evmTJktWrVy9atAjvWghKU1MzIiIiPDycx+OdOnUK73KIKDU1lc/nf/LJJ3gXQnRVVVU3btzIyclJSUkJCQlpbW2VyrTydUSop6cnFoubm5vpdDpCqKmpCe5hA2PB4/E8PT3t7OzgjNYQVFRUmEwmQsjMzMzFxWXnzp2SP0YB5tChQ3p6eiEhIQghLpd75MgRPp/v4eGBd12EQ6FQsBdeXl4UCqWsrMzR0XHs08rXr+PEiRNnzpyZkZGBEBKLxRkZGXBFGoxaZ2ent7e3mZlZXFwciUTCuxwZ0NzcrKqqCinY3/79+9esWcNkMplMpqKiop2d3fCbSsun8vJyHo8nrbVx5OuIECEUFRUVEhLy6tWrx48fs9nslStX4l0REaWlpV2+fDk/P7+7uzskJMTT09PLywvvoggnPDz87t27JiYm69evRwgxGIzo6Gi8iyKczZs3v3nzxsjI6OXLl0lJSTt27MC7IiJyc3OTvA4NDZ07d66lpSWO9RDTtWvXkpOTbW1tOzo6zpw5ExoaKrnhY4zk7vEJhFBeXl5aWpqGhoa/v7+mpibe5RBRUVHRgwcPJG/t7Ozs7OxwrIeYbt26VV1dLXmrra29dOlSHOshpoqKips3b9bV1WlpaTGZTFtbW7wrIrpz587Nnz8frtr0x2azr1y5UlVVRaVSnZyc5s2bJ62Z5TEIAQAAAAk4WQ8AAECuQRACAACQaxCEAAAA5BoEIQAAALkGQQgAAECuQRACAACQaxCEAIxDV69exbsEAGQGBCEA401hYeHGjRvxrgIAmQFBCMB4k5WV5eLigncVAMgMCEIAxo/nz58XFhamp6fr6+sXFhby+Xy8KwJABkAQAjB+sFis58+f37lzx8TEpKampru7G++KAJAB0GsUgHGlsLDQy8uroaEB70IAkBlwRAjAuJKdnQ2rbAIwIhCEAIwrOTk5EIQAjAgEIQDjR09Pz19//YUF4Z07d/AuBwDZAEEIwPjR0NDQ09Njbm7e3t7+7NkzvMsBQDYo4F0AAEBq6HS6vb39+fPnORxOUFAQ3uUAIBvgrlEAxpuuri4qlYp3FQDIDAhCAAAAcg2uEQIAAJBrEIQAAADkGgQhAAAAuQZBCAAAQK5BEAIAAJBrEIQAAADkGgQhAAAAuQZBCAAAQK79PzyY5uPI5fx0AAAAAElFTkSuQmCC", "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" ], "text/html": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "u0, T, n = 0., 5., 20\n", "f(t,u) = u - t^2 + 1\n", "f_t(t,u) = -2*t\n", "f_u(t,u) = 1\n", "ϕ(t, u, h) = f(t,u) + (h/2)*( f_t(t,u) + f_u(t,u) * f(t,u) )\n", "\n", "u1 = oneStep( u0, (t,u,h)->f(t,u), T, n )\n", "u2 = oneStep( u0, ϕ, T, n )\n", "\n", "u_exact(t) = (t+1)^2 - exp(t)\n", "\n", "plot(u_exact, 0, T, \n", " label=\"exact\",\n", " xlabel=L\"t\", ylabel=L\"u(t)\")\n", "\n", "scatter!( 0:T/n:T, u1, \n", " label=\"Euler (n=$n)\")\n", "\n", "scatter!( 0:T/n:T, u2, \n", " label=\"Taylor order 2 (n=$n)\" )" ] }, { "cell_type": "markdown", "id": "341f836c", "metadata": {}, "source": [ "Here's the error as a function of $n$:" ] }, { "cell_type": "code", "execution_count": 4, "id": "6dba1cf0", "metadata": {}, "outputs": [ { "data": { "image/png": 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"\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": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 500\n", "errs1 = zeros(N)\n", "errs2 = zeros(N)\n", "for n ∈ 1:N\n", " u1 = oneStep( u0, (t,u,h)->f(t,u), T, n )\n", " u2 = oneStep( u0, ϕ, T, n )\n", " mesh = 0:T/n:T\n", " errs1[n] = maximum( @. abs( u1 - u_exact(mesh) ) )\n", " errs2[n] = maximum( @. abs( u2 - u_exact(mesh) ) )\n", "end\n", "\n", "scatter( errs1, xaxis=:log, yaxis=:log, xlabel=L\"n\", label=\"Euler's method\", lw=3, title=\"Error\" )\n", "plot!( 500*(1:N).^(-1), label=L\"n^{-1}\", linestyle=:dash )\n", "\n", "scatter!( errs2, xaxis=:log, yaxis=:log, xlabel=L\"n\", label=\"Taylor order 2\", lw=3 )\n", "plot!( 1000*(1:N).^(-2), label=L\"n^{-2}\", linestyle=:dash )" ] }, { "cell_type": "markdown", "id": "2255a345", "metadata": {}, "source": [ ":::\n", "\n", "However, the formulae for the higher order derivatives of $f$ become more complicated and we don't want to compute the derivatives of $f$ explicitly.\n", "\n", "## Runge-Kutta Methods\n", "\n", "Taylor theorem in multiple variables, Runge-Kutta of order $2$: @Burden section 5.4 \n", "\n", "*Midpoint method* is often written as a *multi-stage method*:\n", "\n", "\\begin{align}\n", " k_1 &:= h f(t_j, u_j) \\nonumber\\\\\n", " k_2 &:= h f(t_j + \\tfrac12 h, u_j + \\tfrac12 k_1) \\nonumber\\\\\n", " u_{j+1} &:= u_j + k_2 \n", " \\tag{Midpoint}\n", "\\end{align}\n", "\n", "::: {#exm-2}\n", "\n", "We return to @exm-1 but now also implement the midpoint method" ] }, { "cell_type": "code", "execution_count": 5, "id": "4f2bcf9c", "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", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "n = 20\n", "ϕ2(t, u, h) = f(t+h/2,u+(h/2)*f(t,u))\n", "\n", "u1 = oneStep( u0, (t,u,h)->f(t,u), T, n )\n", "u2 = oneStep( u0, ϕ, T, n )\n", "u3 = oneStep( u0, ϕ2, T, n )\n", "\n", "plot(u_exact, 0, T, \n", " label=\"exact\",\n", " xlabel=L\"t\", ylabel=L\"u(t)\")\n", "\n", "scatter!( 0:T/n:T, u1, \n", " label=\"Euler (n=$n)\")\n", "\n", "scatter!( 0:T/n:T, u2, \n", " label=\"Taylor order 2 (n=$n)\" )\n", "\n", "scatter!( 0:T/n:T, u3, \n", " label=\"Midpoint (n=$n)\" )" ] }, { "cell_type": "markdown", "id": "8ea99e42", "metadata": {}, "source": [ "Here's a plot of the errors:" ] }, { "cell_type": "code", "execution_count": 6, "id": "bd7cc0f5", "metadata": {}, "outputs": [ { "data": { "image/png": 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"\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": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "N = 500\n", "errs3 = zeros(N)\n", "for n ∈ 1:N\n", " u3 = oneStep( u0, ϕ2, T, n )\n", " mesh = 0:T/n:T\n", " errs3[n] = maximum( @. abs( u3 - u_exact(mesh) ) )\n", "end\n", "\n", "scatter( errs1, xaxis=:log, yaxis=:log, xlabel=L\"n\", label=\"Euler's method\", lw=3, title=\"Error\" )\n", "plot!( 500*(1:N).^(-1), label=L\"n^{-1}\", linestyle=:dash )\n", "\n", "scatter!( errs2, xaxis=:log, yaxis=:log, xlabel=L\"n\", label=\"Taylor order 2\", lw=3 )\n", "scatter!( errs3, xaxis=:log, yaxis=:log, xlabel=L\"n\", label=\"Midpoint\", lw=3 )\n", "plot!( 1000*(1:N).^(-2), label=L\"n^{-2}\", linestyle=:dash )" ] }, { "cell_type": "markdown", "id": "cc1ca527", "metadata": {}, "source": [ "Here, we can see the order of accuracy is indeed $p=2$.\n", "\n", ":::\n", "\n", "::: {#def-RK}\n", "\n", "A general $s$-stage Runge-Kutta method is of the form:\n", "\n", "\\begin{align}\n", " k_1 &= h f(t_j, u_j) \\nonumber\\\\\n", " k_2 &= h f(t_j + c_1 h, u_j + a_{11} k_1) \\nonumber \\\\\n", " k_3 &= h f(t_j + c_2 h, u_j + a_{21} k_1 + a_{22} k_2) \\nonumber\\\\\n", " \\vdots& \\nonumber\\\\\n", " k_s &= h f(t_j + c_{s-1} h, u_j + a_{s-1,1} k_1 + \\dots + a_{s-1,s-1} k_{s-1}) \\nonumber\\\\\n", " %\n", " u_{j+1} &= u_{j} + b_1 k_1 + b_2 k_2 + \\dots + b_s k_s. \\nonumber\n", "\\end{align}\n", "\n", "This Runge-Kutta method is conveniently written as the *Butcher tableau*: \n", "\n", "\\begin{array}\n", "{c|cccc}\n", "0\\\\\n", "c_1 & a_{11} \\\\\n", "c_2 & a_{21} & a_{22} \\\\\n", "\\vdots & \\vdots& \\vdots & \\ddots& \\\\\n", "c_{s-1} & a_{s-1,1}& a_{s-1,2} & \\cdots & a_{s-1,s-1} \\\\\n", "\\hline\n", "& b_1 & b_2 & \\dots & b_{s-1} & b_s \n", "\\end{array}\n", "\n", ":::\n", "\n", "::: {#exr-3}\n", "\n", "Write down the Butcher tableau for *(i)* Euler's method, *(ii)* Midpoint method, and the following $2$-stage, order $2$ methods:\n", "\n", "*(iii)*:\n", "\\begin{align}\n", " u_{j+1} &= u_j + \\frac{h}{2} \\left[ \n", " f(t_j, u_j) + f\\big( t_{j+1}, u_j + h f(t_j, u_j) \\big)\n", " \\right]\n", "\\end{align}\n", "\n", "*(iv)*:\n", "\\begin{align}\n", " u_{j+1} &= u_j + \\frac{h}{4} \\left[ \n", " f(t_j, u_j) + 3 f\\big( t_{j} + \\tfrac{2}{3} h, u_j + \\tfrac{2}{3} h f(t_j, u_j) \\big)\n", " \\right]\n", "\\end{align}\n", "\n", ":::\n", "\n", "::: {#exr-4}\n", "\n", "Show that the Modified Euler method has order of accuracy $2$.\n", "\n", ":::" ] }, { "cell_type": "markdown", "id": "b94ee96d", "metadata": {}, "source": [ ":::{.callout-tip}\n", "# TO-DO\n", "\n", "* Assignment 2: Due next Wednesday,\n", "* Read: @Burden sections 5.1 - 5.4 - please come to the problem class on Wednesday with any questions you have,\n", "* (Sign-up to office hours if you would like)\n", "\n", "Next: Problem class\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 }