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Created March 12, 2014 14:36
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{
"metadata": {
"name": ""
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Demonstration of different limiters for advection\n",
"This notebook shows the effect of limiters on advection for various initial conditions. It includes the results for first and second order methods from Figures 6.1-6.3 of [Finite Volume Methods for Hyperbolic Problems](http://depts.washington.edu/clawpack/book.html), as well as results for WENO methods."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"%matplotlib inline\n",
"import numpy as np\n",
"from clawpack import pyclaw\n",
"from clawpack import riemann\n",
"import matplotlib\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": 10
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The function below sets up an advection simulation with the specified scheme, on the unit interval with periodic boundary conditions. Since output is written at integer times, the solution at each output time has been advected precisely back to its original location. Plotting each frame shows us how the solution is modified by numerical effects as it moves once through the domain."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def setup(scheme='minmod',cfl_max=0.9,IC='gauss_square',mx=100):\n",
" if 'weno' in scheme:\n",
" solver = pyclaw.SharpClawSolver1D(riemann.advection_1D)\n",
" else:\n",
" solver = pyclaw.ClawSolver1D(riemann.advection_1D)\n",
"\n",
" solver.bc_lower[0] = pyclaw.BC.periodic\n",
" solver.bc_upper[0] = pyclaw.BC.periodic\n",
" \n",
" if scheme in ('minmod','superbee','MC','vanleer'):\n",
" solver.limiters = getattr(pyclaw.limiters.tvd,scheme)\n",
" #elif scheme == 'CT':\n",
" #solver.limiters = pyclaw.limiters.tvd.cada_torrilhon_limiter\n",
" elif scheme == 'Lax-Wendroff':\n",
" solver.limiters = 0\n",
" elif scheme == 'first-order':\n",
" solver.order = 1\n",
" elif 'weno' in scheme:\n",
" solver.weno_order = int(scheme[4:]) #weno5, weno7, ...\n",
" else:\n",
" raise Exception('Unrecognized limiter')\n",
"\n",
" solver.cfl_max = cfl_max\n",
" solver.cfl_desired = cfl_max*0.9\n",
"\n",
" x = pyclaw.Dimension('x',0.0,1.0,mx)\n",
" domain = pyclaw.Domain(x)\n",
" num_eqn = 1\n",
" state = pyclaw.State(domain,num_eqn)\n",
" state.problem_data['u']=1.\n",
"\n",
" grid = state.grid\n",
" xc = grid.x.centers\n",
" if IC=='gauss_square':\n",
" beta=200.; x0=0.3\n",
" state.q[0,:] = np.exp(-beta * (xc-x0)**2) + (xc>0.6)*(xc<0.8)\n",
" elif IC=='wavepacket':\n",
" beta=100.; x0=0.5\n",
" state.q[0,:] = np.exp(-beta * (xc-x0)**2) * np.sin(80.*xc)\n",
" else:\n",
" raise Exception('Unrecognized initial condition.')\n",
"\n",
" claw = pyclaw.Controller()\n",
" claw.solution = pyclaw.Solution(state,domain)\n",
" claw.solver = solver\n",
" claw.keep_copy = True\n",
" claw.output_format = None\n",
"\n",
" claw.tfinal =10.0\n",
" return claw"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 11
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Gaussian and square wave"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"results = []\n",
"schemes = ('first-order','Lax-Wendroff','minmod','superbee','MC','vanleer','weno5','weno7','weno9')\n",
"for scheme in schemes:\n",
" claw = setup(scheme=scheme)\n",
" claw.run()\n",
" results.append(claw.frames)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 23
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"def animate(results,ymin=-0.1):\n",
" fig = plt.figure(figsize=(10,6))\n",
"\n",
" N = len(results)\n",
" n = int(np.ceil(np.sqrt(N)))\n",
" axes = []\n",
" gs1 = matplotlib.gridspec.GridSpec(n, n)\n",
" gs1.update(wspace=0.,hspace=0.)\n",
"\n",
" for i in range(n):\n",
" for j in range(n):\n",
" k = n*i + j\n",
" if k<N:\n",
" axes.append(plt.subplot(gs1[i,j]));\n",
" if j>0:\n",
" axes[-1].yaxis.set_ticklabels(())\n",
"\n",
" lines = [0]*len(schemes)\n",
" for i in range(len(lines)):\n",
" lines[i], = axes[i].plot([], [], lw=2)\n",
"\n",
" for i,ax in enumerate(axes):\n",
" ax.set_xlim(0,1); ax.set_ylim(ymin,1.3)\n",
" #ax.legend( [schemes[i] ] )\n",
" ax.set_title(schemes[i], x = 0.5, y=0.85 )\n",
" \n",
" xc = results[0][0].p_centers[0]\n",
"\n",
" def fplot(frame_number):\n",
" for i, line in enumerate(lines):\n",
" line.set_data(xc,results[i][frame_number].q[0,:])\n",
" return lines,\n",
"\n",
" return matplotlib.animation.FuncAnimation(fig, fplot, frames=len(claw.frames), interval=30)\n",
"\n",
"animate(results)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"html": [
"\n",
"<script language=\"javascript\">\n",
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" this.slider_id = slider_id;\n",
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" 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",
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"\n",
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"\n",
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"\n",
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" }\n",
"\n",
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" {\n",
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"\"\n",
"\n",
"\n",
" /* set a timeout to make sure all the above elements are created before\n",
" the object is initialized. */\n",
" setTimeout(function() {\n",
" anim8c51b58952bd39a9 = new Animation(frames, img_id, slider_id, frame_id, 29, loop_select_id);\n",
" }, 0);\n",
" })()\n",
"</script>\n"
],
"metadata": {},
"output_type": "pyout",
"prompt_number": 24,
"text": [
"<matplotlib.animation.FuncAnimation at 0x10c8d5a90>"
]
}
],
"prompt_number": 24
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It's easy to see the differences in dissipation and dispersion of the various schemes. Of course, these properties also depend on the CFL number. Try changing the CFL number in the call to `setup()` above and see how the results change."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Wavepacket\n",
"To see how the limiters affect intermediate frequencies, let's advect a wavepacket."
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"results = []\n",
"\n",
"for scheme in schemes:\n",
" claw = setup(scheme=scheme,IC='wavepacket',mx=300)\n",
" claw.run()\n",
" results.append(claw.frames)"
],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 25
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"animate(results,ymin=-1.2)"
],
"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",
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" }\n",
"\n",
" Animation.prototype.anim_step_reverse = function()\n",
" {\n",
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" this.set_frame(this.current_frame);\n",
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" var loop_state = this.get_loop_state();\n",
" if(loop_state == \"loop\"){\n",
" this.last_frame();\n",
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" 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",
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" <img id=\"_anim_imgcc46efa69af80fc9\" style=\"width:Nonepx\"> \n",
" <br>\n",
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" <button onclick=\"animcc46efa69af80fc9.faster()\">+</button>\n",
" <form action=\"#n\" name=\"_anim_loop_selectcc46efa69af80fc9\" class=\"anim_control\">\n",
" <input id=\"_frame_nocc46efa69af80fc9\" type=\"textbox\" size=\"1\" onchange=\"animcc46efa69af80fc9.set_frame(parseInt(this.value));\" onpaste=\"this.onchange();\" oninput=\"this.onchange();\"></input>\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",
"</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_imgcc46efa69af80fc9\";\n",
" var slider_id = \"_anim_slidercc46efa69af80fc9\";\n",
" var frame_id = \"_frame_nocc46efa69af80fc9\"\n",
" var loop_select_id = \"_anim_loop_selectcc46efa69af80fc9\";\n",
" var frames = new Array(0);\n",
" \n",
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