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July 2, 2014 17:11
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{ | |
"metadata": { | |
"name": "", | |
"signature": "sha256:4f433c47c93b066dd71340745fbbb90411a1028947e26752c7102b9cbe55a900" | |
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"worksheets": [ | |
{ | |
"cells": [ | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"# Multigrid\n", | |
"\n", | |
"Multigrid is one of the great algorithms for numerically solving PDEs. It is also one of the few essentially optimal algorithms for solving a linear system of equations, since it computes the solution of an $N\\times N$ system in ${\\mathcal O}(N\\log(N))$ -- or even just ${\\mathcal O}(N)$ -- operations. \n", | |
"\n", | |
"This notebook is meant to accompany a reading of [Section 4.6 of Randall LeVeque's text on finite difference methods](http://0-epubs.siam.org.library.kaust.edu.sa/doi/abs/10.1137/1.9780898717839.ch4). Other good resources are cited there." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"collapsed": false, | |
"input": [ | |
"%matplotlib inline\n", | |
"import numpy as np\n", | |
"import matplotlib.pyplot as plt\n", | |
"from matplotlib import animation\n", | |
"from clawpack.visclaw.JSAnimation import IPython_display" | |
], | |
"language": "python", | |
"metadata": {}, | |
"outputs": [], | |
"prompt_number": 1 | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"## A two-point boundary value problem\n", | |
"Let's use Jacobi's method to solve the one-dimensional boundary value problem \n", | |
"\n", | |
"\\begin{align}\n", | |
"u''(x) & = f(x) & 0<x<1\n", | |
"\\end{align}\n", | |
"with boundary conditions $u(0) = 1, u(1) = 3$ where\n", | |
"\\begin{align}\n", | |
"f(x) & = -20 + \\frac{1}{2} \\phi''(x) \\cos(\\phi(x)) - \\frac{1}{2} (\\phi'(x))^2\\sin(\\phi(x)) \\\\ \\\\\n", | |
"\\phi(x) & = 20\\pi x^3.\n", | |
"\\end{align}\n", | |
"This problem appears on page 103 of the text. In the box below, we set up the grid, the right hand side, and the boundary values." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"collapsed": false, | |
"input": [ | |
"m=2**8-1 # Number of grid points\n", | |
"h=1./(m+1) # Mesh width\n", | |
"x=np.linspace(0,1,m+2); x=x[1:-1] # grid\n", | |
"alpha=1.; beta=3. # boundary values\n", | |
"phi = lambda x: 20.* np.pi * x**3\n", | |
"f = lambda x: -20 + 0.5*120*np.pi*x * np.cos(phi(x)) - 0.5*(60*np.pi*x**2)**2 * np.sin(phi(x)) # RHS\n", | |
"plt.plot(x,f(x)); plt.xlabel('x'); plt.ylabel('f(x)')" | |
], | |
"language": "python", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"metadata": {}, | |
"output_type": "pyout", | |
"prompt_number": 2, | |
"text": [ | |
"<matplotlib.text.Text at 0x108607810>" | |
] | |
}, | |
{ | |
"metadata": {}, | |
"output_type": "display_data", | |
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| |
"text": [ | |
"<matplotlib.figure.Figure at 0x108601ed0>" | |
] | |
} | |
], | |
"prompt_number": 2 | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"## Jacobi's Method\n", | |
"A centered-difference method for this problem gives us the system of equations\n", | |
"\n", | |
"\\begin{align}\n", | |
"\\frac{U_{i-1} - 2 U_i + U_{i+1}}{h^2} & = f(x_i).\n", | |
"\\end{align}\n", | |
"\n", | |
"Solving for $U_i$ gives\n", | |
"\\begin{align}\n", | |
"U_i & = \\frac{1}{2} ( U_{i+1} + U_{i-1} ) - \\frac{h^2}{2} f(x_i).\n", | |
"\\end{align}\n", | |
"\n", | |
"Jacobi's method consists of using the equation above as a fixed-point iteration:\n", | |
"\n", | |
"\\begin{align}\n", | |
"U_i^{[k+1]} & = \\frac{1}{2} ( U_{i+1}^{[k]} + U_{i-1}^{[k]} ) - \\frac{h^2}{2} f(x_i).\n", | |
"\\end{align}" | |
] | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"In the cell below, implement Jacobi's method for the BVP. Don't forget about the boundary conditions. Take 100 iterations with it and plot the result." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"collapsed": true, | |
"input": [ | |
"U=np.linspace(alpha,beta,m) # Just use a straight line as initial guess\n", | |
"UU=[U]\n", | |
"F=0.5*h**2*f(x) # Construct the RHS\n", | |
"F[0]-=alpha/2.; F[-1]-=beta/2. # boundary conditions\n", | |
"e=np.ones(m-1)\n", | |
"G=0.5*(np.diag(e,-1)+np.diag(e,1)) # Jacobi matrix (average neighbors)\n", | |
"\n", | |
"for i in range(100):\n", | |
" U=np.dot(G,U) - F\n", | |
" UU.append(U)" | |
], | |
"language": "python", | |
"metadata": {}, | |
"outputs": [], | |
"prompt_number": 3 | |
}, | |
{ | |
"cell_type": "markdown", | |
"metadata": {}, | |
"source": [ | |
"If you can, try storing the solution after each Jacobi iteration in a list, and make an animated plot of the solution after each iteration." | |
] | |
}, | |
{ | |
"cell_type": "code", | |
"collapsed": false, | |
"input": [ | |
"xf = np.linspace(0,1,1000); \n", | |
"uf = 1.+12.*xf-10.*xf**2 + 0.5*np.sin(phi(xf)) # Exact solution on fine grid\n", | |
"u = 1.+12.*x-10.*x**2 + 0.5*np.sin(phi(x)) # Exact solution on computational grid\n", | |
"\n", | |
"fig = plt.figure()\n", | |
"ax = plt.axes(xlim=(0, 1), ylim=(-3, 6))\n", | |
"ax.plot(xf,uf)\n", | |
"line1, = ax.plot([],[],'-')\n", | |
"line2, = ax.plot([],[],'-o')\n", | |
"plt.legend(['Exact','Jacobi','Error'],loc=2)\n", | |
"\n", | |
"def fplot(i):\n", | |
" line1.set_data(x,UU[i])\n", | |
" line2.set_data(x,UU[i]-u)\n", | |
" ax.set_title('# of iterations: '+str(i))\n", | |
" return line1, \n", | |
"\n", | |
"animation.FuncAnimation(fig, fplot, frames=range(len(UU)))" | |
], | |
"language": "python", | |
"metadata": {}, | |
"outputs": [ | |
{ | |
"html": [ | |
"\n", | |
"<script language=\"javascript\">\n", | |
" /* Define the Animation class */\n", | |
" function Animation(frames, img_id, slider_id, frame_id, interval, loop_select_id){\n", | |
" this.img_id = img_id;\n", | |
" this.slider_id = slider_id;\n", | |
" this.frame_id = frame_id;\n", | |
" this.loop_select_id = loop_select_id;\n", | |
" this.interval = interval;\n", | |
" this.current_frame = 0;\n", | |
" this.direction = 0;\n", | |
" this.timer = null;\n", | |
" this.frames = new Array(frames.length);\n", | |
"\n", | |
" for (var i=0; i<frames.length; i++)\n", | |
" {\n", | |
" this.frames[i] = new Image();\n", | |
" this.frames[i].src = frames[i];\n", | |
" }\n", | |
" document.getElementById(this.slider_id).max = this.frames.length - 1;\n", | |
" this.set_frame(this.current_frame);\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.get_loop_state = function(){\n", | |
" var button_group = document[this.loop_select_id].state;\n", | |
" for (var i = 0; i < button_group.length; i++) {\n", | |
" var button = button_group[i];\n", | |
" if (button.checked) {\n", | |
" return button.value;\n", | |
" }\n", | |
" }\n", | |
" return undefined;\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.set_frame = function(frame){\n", | |
" this.current_frame = frame;\n", | |
" document.getElementById(this.img_id).src = this.frames[this.current_frame].src;\n", | |
" document.getElementById(this.slider_id).value = this.current_frame;\n", | |
" document.getElementById(this.frame_id).value = this.current_frame;\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.next_frame = function()\n", | |
" {\n", | |
" this.set_frame(Math.min(this.frames.length - 1, this.current_frame + 1));\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.previous_frame = function()\n", | |
" {\n", | |
" this.set_frame(Math.max(0, this.current_frame - 1));\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.first_frame = function()\n", | |
" {\n", | |
" this.set_frame(0);\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.last_frame = function()\n", | |
" {\n", | |
" this.set_frame(this.frames.length - 1);\n", | |
" } \n", | |
"\n", | |
" Animation.prototype.slower = function()\n", | |
" {\n", | |
" this.interval /= 0.7;\n", | |
" if(this.direction > 0){this.play_animation();}\n", | |
" else if(this.direction < 0){this.reverse_animation();}\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.faster = function()\n", | |
" {\n", | |
" this.interval *= 0.7;\n", | |
" if(this.direction > 0){this.play_animation();}\n", | |
" else if(this.direction < 0){this.reverse_animation();}\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.anim_step_forward = function()\n", | |
" {\n", | |
" this.current_frame += 1;\n", | |
" if(this.current_frame < this.frames.length){\n", | |
" this.set_frame(this.current_frame);\n", | |
" }else{\n", | |
" var loop_state = this.get_loop_state();\n", | |
" if(loop_state == \"loop\"){\n", | |
" this.first_frame();\n", | |
" }else if(loop_state == \"reflect\"){\n", | |
" this.last_frame();\n", | |
" this.reverse_animation();\n", | |
" }else{\n", | |
" this.pause_animation();\n", | |
" this.last_frame();\n", | |
" }\n", | |
" }\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.anim_step_reverse = function()\n", | |
" {\n", | |
" this.current_frame -= 1;\n", | |
" if(this.current_frame >= 0){\n", | |
" this.set_frame(this.current_frame);\n", | |
" }else{\n", | |
" var loop_state = this.get_loop_state();\n", | |
" if(loop_state == \"loop\"){\n", | |
" this.last_frame();\n", | |
" }else if(loop_state == \"reflect\"){\n", | |
" this.first_frame();\n", | |
" this.play_animation();\n", | |
" }else{\n", | |
" this.pause_animation();\n", | |
" this.first_frame();\n", | |
" }\n", | |
" }\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.pause_animation = function()\n", | |
" {\n", | |
" this.direction = 0;\n", | |
" if (this.timer){\n", | |
" clearInterval(this.timer);\n", | |
" this.timer = null;\n", | |
" }\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.play_animation = function()\n", | |
" {\n", | |
" this.pause_animation();\n", | |
" this.direction = 1;\n", | |
" var t = this;\n", | |
" if (!this.timer) this.timer = setInterval(function(){t.anim_step_forward();}, this.interval);\n", | |
" }\n", | |
"\n", | |
" Animation.prototype.reverse_animation = function()\n", | |
" {\n", | |
" this.pause_animation();\n", | |
" this.direction = -1;\n", | |
" var t = this;\n", | |
" if (!this.timer) this.timer = setInterval(function(){t.anim_step_reverse();}, this.interval);\n", | |
" }\n", | |
"</script>\n", | |
"\n", | |
"<div class=\"animation\" align=\"center\">\n", | |
" <img id=\"_anim_imgccd946df81dc5209\" style=\"width:Nonepx\">\n", | |
" <br>\n", | |
" <input id=\"_anim_sliderccd946df81dc5209\" type=\"range\" style=\"width:350px\" name=\"points\" min=\"0\" max=\"1\" step=\"1\" value=\"0\" onchange=\"animccd946df81dc5209.set_frame(parseInt(this.value));\"></input>\n", | |
" <br>\n", | |
" \n", | |
" <button onclick=\"animccd946df81dc5209.slower()\">–</button>\n", | |
" <button onclick=\"animccd946df81dc5209.first_frame()\"><img class=\"anim_icon\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAUCAQAAAAngNWGAAAAAXNSR0IArs4c6QAAAAJiS0dEAP+H\n", | |
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"rV+/blxKMZUnLgAAAABJRU5ErkJggg==\n", | |
"\"></button>\n", | |
" <button onclick=\"animccd946df81dc5209.previous_frame()\"><img class=\"anim_icon\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAUCAQAAAAngNWGAAAAAXNSR0IArs4c6QAAAAJiS0dEAP+H\n", | |
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"h8FrxQwtPU99HrwBXsnAwMDAsJiBgYGBoZ1xmKYqALHhMpn1o7igAAAAAElFTkSuQmCC\n", | |
"\"></button>\n", | |
" <button onclick=\"animccd946df81dc5209.reverse_animation()\"><img class=\"anim_icon\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAUCAQAAAAngNWGAAAAAXNSR0IArs4c6QAAAAJiS0dEAP+H\n", | |
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"RK5CYII=\n", | |
"\"></button>\n", | |
" <button onclick=\"animccd946df81dc5209.pause_animation()\"><img class=\"anim_icon\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAUCAQAAAAngNWGAAAAAXNSR0IArs4c6QAAAAJiS0dEAP+H\n", | |
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"+MfwH64Qnnq0zr9fyfLrPzP3eQYGBobvk5x4GX4xMIij23gdib0cRWYHiVmAAQDK5ircshCbHQAA\n", | |
"AABJRU5ErkJggg==\n", | |
"\"></button>\n", | |
" <button onclick=\"animccd946df81dc5209.play_animation()\"><img class=\"anim_icon\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAUCAQAAAAngNWGAAAAAXNSR0IArs4c6QAAAAJiS0dEAP+H\n", | |
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"Rpcle+Mf8wvgJ16zo/4BtQAAAABJRU5ErkJggg==\n", | |
"\"></button>\n", | |
" <button onclick=\"animccd946df81dc5209.next_frame()\"><img class=\"anim_icon\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAUCAQAAAAngNWGAAAAAXNSR0IArs4c6QAAAAJiS0dEAP+H\n", | |
"j8y/AAAACXBIWXMAAAsTAAALEwEAmpwYAAAAB3RJTUUH3QURCAkd/uac8wAAAMhJREFUKM9jYBie\n", | |
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"qNIwUMDAwMDAsADKS2NkGL4AAIARMlfNIfZMAAAAAElFTkSuQmCC\n", | |
"\"></button>\n", | |
" <button onclick=\"animccd946df81dc5209.last_frame()\"><img class=\"anim_icon\" src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABQAAAAUCAQAAAAngNWGAAAAAXNSR0IArs4c6QAAAAJiS0dEAP+H\n", | |
"j8y/AAAACXBIWXMAAAsTAAALEwEAmpwYAAAAB3RJTUUH3QURCAknOOpFQQAAAS9JREFUKM/dkrEv\n", | |
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"SaXq9Wd4P03/ht5y98197/u9XA4aK4rAWw3lgWddZ3S+/G9mEovtAB8AHE4pgTQAx8PbJweRmsq6\n", | |
"GimmNpxaNYXVzMNNCI6A2figimwCGACK786zuWgh3qcsKf/w0pM4X0m/doNVFVzVGlEQsdRj193V\n", | |
"xEWpH0RsdRu+zi3tVMqCAsDShoiYqiSV4OouVDFEqS9Pbiyg7vV62lpQ2BJ4Gg0meg0MbNpkYG/e\n", | |
"+540NNFyrE1a8qHk5BaAjfnrzUaHfAWImVrLIXbgnx4/9X06s35cweWsVACa3a24PVp0X+rPv1aH\n", | |
"FnSONdiL8Qci0lzwpOM5sQAAAABJRU5ErkJggg==\n", | |
"\"></button>\n", | |
" <button onclick=\"animccd946df81dc5209.faster()\">+</button>\n", | |
" <form action=\"#n\" name=\"_anim_loop_selectccd946df81dc5209\" class=\"anim_control\">\n", | |
"\n", | |
" <input id=\"_frame_noccd946df81dc5209\" type=\"textbox\" size=\"1\" onchange=\"animccd946df81dc5209.set_frame(parseInt(this.value));\" onpaste=\"this.onchange();\" oninput=\"this.onchange();\"></input>\n", | |
" \n", | |
" <input type=\"radio\" name=\"state\" value=\"once\" > Once </input>\n", | |
" <input type=\"radio\" name=\"state\" value=\"loop\" checked> Loop </input>\n", | |
" <input type=\"radio\" name=\"state\" value=\"reflect\" > Reflect </input>\n", | |
" </form>\n", | |
"\n", | |
"</div>\n", | |
"\n", | |
"\n", | |
"<script language=\"javascript\">\n", | |
" /* Instantiate the Animation class. */\n", | |
" /* The IDs given should match those used in the template above. */\n", | |
" (function() {\n", | |
" var img_id = \"_anim_imgccd946df81dc5209\";\n", | |
" var slider_id = \"_anim_sliderccd946df81dc5209\";\n", | |
" var frame_id = \"_frame_noccd946df81dc5209\"\n", | |
" var loop_select_id = \"_anim_loop_selectccd946df81dc5209\";\n", | |
" var frames = new Array(0);\n", | |
" \n", | |
" frames[0] = \"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbAAAAEgCAYAAADVKCZpAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\\\n", | |
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