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ketch/Guide_to_limiters.ipynb

Last active August 29, 2015 14:21
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Guide_to_limiters.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"###### Content provided under a Creative Commons Attribution license, CC-BY 4.0; code under MIT License. (c)2014 [David I. Ketcheson](http://davidketcheson.info)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#An illustrated guide to limiters\n",
"## Or: how to interpolate non-smooth data without creating wiggles"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Many interesting wave phenomena -- like fluid dynamics, lasers, and water waves -- are described by nonlinear hyperbolic partial differential equations. The solutions of these problems are discontinuous. So-called **limiters** (sometimes referred to as *slope limiters* or *flux limiters* are one of the key ingredients in approximating these discontinuous solutions."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Table of contents\n",
"- [Motivation: interpolation and wiggles](#Interpolation-and-wiggles)\n",
"- [The simplest limiter: Minmod](#The-simplest-limiter:-Minmod)\n",
"- [Other TVD limiters](#TVD-limiters)\n",
"- [WENO](#Higher-order-interpolation:-WENO)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import matplotlib.pyplot as plt\n",
"import numpy as np\n",
"import matplotlib\n",
"matplotlib.rcParams.update({'font.size': 22})"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import mpld3 # Skip this cell if you don't have mpld3 installed\n",
"mpld3.enable_notebook() # or just go and do it now: pip install mpld3\n",
" # This allows you to zoom and pan the plots."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##Interpolation and wiggles"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Suppose you're given a set of data samples:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
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" });\n",
" });\n",
"}else{\n",
" // require.js not available: dynamically load d3 & mpld3\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/d3.v3.min.js\", function(){\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
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" })\n",
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],
"text/plain": [
"<matplotlib.figure.Figure at 0x10886ec10>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"k = 5\n",
"x=np.arange(-k+1,k)\n",
"y=np.sin(x/2.)+1.\n",
"width = 12\n",
"size = (width,4)\n",
"plt.figure(figsize=size)\n",
"plt.plot(x,y,'or',markersize=10,alpha=0.5)\n",
"plt.axis( (-k, k, -0.1, 2.1) );"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now what you really want to know is, what is the state of the system at the points halfway between your samples? And to figure that out, you need to guess at what's going on in the times in-between those samples. The simplest approximation would be to assume that the system just jumps from one value to the next somewhere in-between:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
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" }(mpld3);\n",
"}else if(typeof define === \"function\" && define.amd){\n",
" // require.js is available: use it to load d3/mpld3\n",
" require.config({paths: {d3: \"https://mpld3.github.io/js/d3.v3.min\"}});\n",
" require([\"d3\"], function(d3){\n",
" window.d3 = d3;\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
" mpld3.draw_figure(\"fig_el8271744381023524810329107\", {\"axes\": [{\"xlim\": [-5.0, 5.0], \"yscale\": \"linear\", \"axesbg\": \"none\", \"texts\": [{\"v_baseline\": \"auto\", \"h_anchor\": \"middle\", \"color\": \"#000000\", \"text\": \"Piecewise-constant approximation\", \"coordinates\": \"axes\", \"zorder\": 3, \"alpha\": 1, \"fontsize\": 20.0, \"position\": [0.5, 1.0224014336917562], \"rotation\": -0.0, \"id\": \"el827174440841104\"}], \"zoomable\": true, \"images\": [], \"xdomain\": [-5.0, 5.0], \"ylim\": [-0.10000000000000001, 2.1000000000000001], \"paths\": [], \"sharey\": [], \"sharex\": [], \"axesbgalpha\": null, \"axes\": [{\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 7, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", 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" });\n",
" });\n",
"}else{\n",
" // require.js not available: dynamically load d3 & mpld3\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/d3.v3.min.js\", function(){\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
" mpld3.draw_figure(\"fig_el8271744381023524810329107\", {\"axes\": [{\"xlim\": [-5.0, 5.0], \"yscale\": \"linear\", \"axesbg\": \"none\", \"texts\": [{\"v_baseline\": \"auto\", \"h_anchor\": \"middle\", \"color\": \"#000000\", \"text\": \"Piecewise-constant approximation\", \"coordinates\": \"axes\", \"zorder\": 3, \"alpha\": 1, \"fontsize\": 20.0, \"position\": [0.5, 1.0224014336917562], \"rotation\": -0.0, \"id\": \"el827174440841104\"}], \"zoomable\": true, \"images\": [], \"xdomain\": [-5.0, 5.0], \"ylim\": [-0.10000000000000001, 2.1000000000000001], \"paths\": [], \"sharey\": [], \"sharex\": [], \"axesbgalpha\": null, \"axes\": [{\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 7, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 1, \"coordinates\": \"data\", \"dasharray\": \"none\", \"zorder\": 2, \"alpha\": 1, \"xindex\": 0, \"linewidth\": 2, \"data\": \"data01\", \"id\": \"el827174441050576\"}], \"markers\": [{\"edgecolor\": \"#000000\", \"facecolor\": \"#FF0000\", \"edgewidth\": 0.5, \"yindex\": 1, \"coordinates\": \"data\", \"zorder\": 2, \"markerpath\": [[[0.0, 5.0], [1.3260155, 5.0], [2.597899353924267, 4.473168457941209], [3.5355339059327378, 3.5355339059327378], [4.473168457941209, 2.597899353924267], [5.0, 1.3260155], [5.0, 0.0], [5.0, -1.3260155], [4.473168457941209, -2.597899353924267], [3.5355339059327378, -3.5355339059327378], [2.597899353924267, -4.473168457941209], [1.3260155, -5.0], [0.0, -5.0], [-1.3260155, -5.0], [-2.597899353924267, -4.473168457941209], [-3.5355339059327378, -3.5355339059327378], [-4.473168457941209, -2.597899353924267], [-5.0, -1.3260155], [-5.0, 0.0], [-5.0, 1.3260155], [-4.473168457941209, 2.597899353924267], [-3.5355339059327378, 3.5355339059327378], [-2.597899353924267, 4.473168457941209], [-1.3260155, 5.0], [0.0, 5.0]], [\"M\", \"C\", \"C\", \"C\", \"C\", \"C\", \"C\", \"C\", \"C\", \"Z\"]], \"alpha\": 0.5, \"xindex\": 0, \"data\": \"data02\", \"id\": \"el827174440702352pts\"}], \"id\": \"el827174440314064\", \"ydomain\": [-0.10000000000000001, 2.1000000000000001], \"collections\": [], \"xscale\": \"linear\", \"bbox\": [0.125, 0.125, 0.77500000000000002, 0.77500000000000002]}], \"height\": 320.0, \"width\": 960.0, \"plugins\": [{\"type\": \"reset\"}, {\"enabled\": false, \"button\": true, \"type\": \"zoom\"}, {\"enabled\": false, \"button\": true, \"type\": \"boxzoom\"}], \"data\": {\"data02\": [[-4.0, 0.09070257317431829], [-3.0, 0.0025050133959455545], [-2.0, 0.1585290151921035], [-1.0, 0.520574461395797], [0.0, 1.0], [1.0, 1.479425538604203], [2.0, 1.8414709848078965], [3.0, 1.9974949866040546], [4.0, 1.9092974268256817]], \"data01\": [[-4.0, 0.09070257317431829], [-3.991991991991992, 0.09070257317431829], [-3.983983983983984, 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" })\n",
" });\n",
"}\n",
"</script>"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x108881150>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def piecewise_constant_interp(x,y,xx):\n",
" \"From samples (x,y) generate piecewise constant function sampled at points xx.\"\n",
" diff = np.abs(x.reshape(1,-1) - xx.reshape(-1,1)) # Here we use numpy broadcasting.\n",
" closest = np.argmin(diff,axis=1)\n",
" return y[closest]\n",
"\n",
"xx = np.linspace(-k+1,k-1,1000)\n",
"yy = piecewise_constant_interp(x,y,xx)\n",
"plt.figure(figsize=size)\n",
"plt.plot(xx,yy,'-k',lw=2)\n",
"plt.hold(True)\n",
"plt.plot(x,y,'or',markersize=10,alpha=0.5)\n",
"plt.axis( (-k, k, -0.1, 2.1) );\n",
"plt.title('Piecewise-constant approximation',fontsize=20);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For this set of data, you don't really believe that's what's happening, do you? But our goal is to deal with systems that exhibit non-smooth (possibly discontinuous) behavior, so we need to at least admit the possibility of sudden jumps. That's why we won't simply \"connect the dots\" to get a continuous approximation.\n",
"\n",
"Instead, we can try to approximate the slope around each of our sample points. The simplest way to do so is using finite differences. If we let $\\sigma_i$ denote our approximation of the slope at $x_i$, then three common approximations are:\n",
"\n",
"- Forward difference: $\\sigma_i = \\frac{y_{i+1}-y_i}{x_{i+1}-x_i}$\n",
"\n",
"\n",
"- Backward difference: $\\sigma_i = \\frac{y_i - y_{i-1}}{x_i - x_{i-1}}$\n",
"\n",
"\n",
"- Centered difference: $\\sigma_i = \\frac{y_{i+1}-y_{i-1}}{x_{i+1}-x_{i-1}}$\n",
"\n",
"Here's what each of these approximations looks like for our data:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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" }(mpld3);\n",
"}else if(typeof define === \"function\" && define.amd){\n",
" // require.js is available: use it to load d3/mpld3\n",
" require.config({paths: {d3: \"https://mpld3.github.io/js/d3.v3.min\"}});\n",
" require([\"d3\"], function(d3){\n",
" window.d3 = d3;\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
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2.037320933008479], [3.263263263263263, 2.0064230918346238, 1.9742758092049435, 2.0385703744643044], [3.271271271271271, 2.0066946691800402, 1.9735695224399514, 2.0398198159201293], [3.2792792792792795, 2.006966246525457, 1.9728632356749594, 2.0410692573759546], [3.2872872872872874, 2.0072378238708737, 1.9721569489099675, 2.04231869883178], [3.295295295295295, 2.00750940121629, 1.9714506621449754, 2.0435681402876047], [3.303303303303303, 2.0077809785617067, 1.9707443753799836, 2.04481758174343], [3.311311311311311, 2.008052555907123, 1.9700380886149915, 2.046067023199255], [3.3193193193193196, 2.0083241332525397, 1.9693318018499995, 2.04731646465508], [3.3273273273273274, 2.008595710597956, 1.9686255150850076, 2.048565906110905], [3.335335335335335, 2.008867287943373, 1.9679192283200155, 2.0498153475667302], [3.343343343343343, 2.0091388652887896, 1.9672129415550237, 2.051064789022555], [3.351351351351351, 2.009410442634206, 1.9665066547900316, 2.0523142304783804], [3.3593593593593596, 2.0096820199796226, 1.9658003680250398, 2.0535636719342056], [3.3673673673673674, 2.009953597325039, 1.9650940812600477, 2.0548131133900305], [3.3753753753753752, 2.0102251746704556, 1.9643877944950559, 2.0560625548458558], [3.383383383383383, 2.010496752015872, 1.9636815077300638, 2.0573119963016806], [3.391391391391391, 2.0107683293612886, 1.962975220965072, 2.058561437757506], [3.3993993993993996, 2.0110399067067055, 1.9622689342000799, 2.0598108792133307], [3.4074074074074074, 2.011311484052122, 1.9615626474350878, 2.061060320669156], [3.4154154154154153, 2.0115830613975385, 1.960856360670096, 2.062309762124981], [3.423423423423423, 2.011854638742955, 1.9601500739051039, 2.063559203580806], [3.431431431431431, 2.0121262160883715, 1.959443787140112, 2.064808645036631], [3.4394394394394396, 2.012397793433788, 1.95873750037512, 2.0660580864924563], [3.4474474474474475, 2.0126693707792045, 1.9580312136101279, 2.0673075279482815], [3.4554554554554553, 2.0129409481246214, 1.957324926845136, 2.0685569694041064], [3.463463463463463, 2.013212525470038, 1.956618640080144, 2.0698064108599317], [3.471471471471472, 2.0134841028154544, 1.955912353315152, 2.071055852315757], [3.4794794794794797, 2.013755680160871, 1.95520606655016, 2.072305293771582], [3.4874874874874875, 2.0140272575062874, 1.9544997797851682, 2.073554735227407], [3.4954954954954953, 2.014298834851704, 1.9537934930201761, 2.074804176683232], [3.503503503503503, 1.9092974268256817, 1.9092974268256817, 1.9530872062551843], [3.511511511511512, 1.9092974268256817, 1.9092974268256817, 1.9523809194901922], [3.5195195195195197, 1.9092974268256817, 1.9092974268256817, 1.9516746327252001], [3.5275275275275275, 1.9092974268256817, 1.9092974268256817, 1.9509683459602083], [3.5355355355355353, 1.9092974268256817, 1.9092974268256817, 1.9502620591952162], [3.543543543543543, 1.9092974268256817, 1.9092974268256817, 1.9495557724302244], [3.551551551551552, 1.9092974268256817, 1.9092974268256817, 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1.9403740444853286], [3.6556556556556554, 1.9092974268256817, 1.9092974268256817, 1.9396677577203367], [3.6636636636636633, 1.9092974268256817, 1.9092974268256817, 1.9389614709553447], [3.671671671671672, 1.9092974268256817, 1.9092974268256817, 1.9382551841903526], [3.67967967967968, 1.9092974268256817, 1.9092974268256817, 1.9375488974253607], [3.6876876876876876, 1.9092974268256817, 1.9092974268256817, 1.9368426106603687], [3.6956956956956954, 1.9092974268256817, 1.9092974268256817, 1.9361363238953768], [3.7037037037037033, 1.9092974268256817, 1.9092974268256817, 1.9354300371303848], [3.711711711711712, 1.9092974268256817, 1.9092974268256817, 1.9347237503653927], [3.71971971971972, 1.9092974268256817, 1.9092974268256817, 1.9340174636004008], [3.7277277277277276, 1.9092974268256817, 1.9092974268256817, 1.9333111768354088], [3.7357357357357355, 1.9092974268256817, 1.9092974268256817, 1.932604890070417], [3.7437437437437433, 1.9092974268256817, 1.9092974268256817, 1.9318986033054248], 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1.9092974268256817, 1.9142414341806255], [3.951951951951952, 1.9092974268256817, 1.9092974268256817, 1.9135351474156335], [3.95995995995996, 1.9092974268256817, 1.9092974268256817, 1.9128288606506416], [3.967967967967968, 1.9092974268256817, 1.9092974268256817, 1.9121225738856495], [3.9759759759759756, 1.9092974268256817, 1.9092974268256817, 1.9114162871206577], [3.9839839839839843, 1.9092974268256817, 1.9092974268256817, 1.9107100003556656], [3.991991991991992, 1.9092974268256817, 1.9092974268256817, 1.9100037135906736], [4.0, 1.9092974268256817, 1.9092974268256817, 1.9092974268256817]]}, \"id\": \"el827174441360848\"});\n",
" });\n",
" });\n",
"}else{\n",
" // require.js not available: dynamically load d3 & mpld3\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/d3.v3.min.js\", function(){\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
" mpld3.draw_figure(\"fig_el8271744413608481404523649\", {\"axes\": [{\"xlim\": [-4.0, 4.0], \"yscale\": \"linear\", \"axesbg\": \"none\", \"texts\": [{\"v_baseline\": \"auto\", \"h_anchor\": \"middle\", \"color\": \"#000000\", \"text\": \"centered\", \"coordinates\": \"axes\", \"zorder\": 3, \"alpha\": 1, \"fontsize\": 20.0, \"position\": [0.5, 0.90000000000000002], \"rotation\": -0.0, \"id\": \"el827174442921808\"}], \"zoomable\": true, \"images\": [], \"xdomain\": [-4.0, 4.0], \"ylim\": [-0.10000000000000001, 2.1000000000000001], \"paths\": [], \"sharey\": [], \"sharex\": [], \"axesbgalpha\": null, \"axes\": [{\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 9, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 1, \"coordinates\": \"data\", \"dasharray\": \"none\", \"zorder\": 2, \"alpha\": 1, \"xindex\": 0, \"linewidth\": 2, \"data\": \"data01\", \"id\": \"el827174442919568\"}], \"markers\": [{\"edgecolor\": \"#000000\", \"facecolor\": \"#FF0000\", \"edgewidth\": 0.5, \"yindex\": 1, \"coordinates\": \"data\", \"zorder\": 2, \"markerpath\": [[[0.0, 5.0], [1.3260155, 5.0], [2.597899353924267, 4.473168457941209], [3.5355339059327378, 3.5355339059327378], [4.473168457941209, 2.597899353924267], [5.0, 1.3260155], [5.0, 0.0], [5.0, -1.3260155], [4.473168457941209, -2.597899353924267], [3.5355339059327378, -3.5355339059327378], [2.597899353924267, -4.473168457941209], [1.3260155, -5.0], [0.0, -5.0], [-1.3260155, -5.0], [-2.597899353924267, -4.473168457941209], [-3.5355339059327378, -3.5355339059327378], [-4.473168457941209, -2.597899353924267], [-5.0, -1.3260155], [-5.0, 0.0], [-5.0, 1.3260155], [-4.473168457941209, 2.597899353924267], [-3.5355339059327378, 3.5355339059327378], [-2.597899353924267, 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\"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 9, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 2, \"coordinates\": \"data\", \"dasharray\": \"none\", \"zorder\": 2, \"alpha\": 1, \"xindex\": 0, \"linewidth\": 2, \"data\": \"data01\", \"id\": \"el827174442969104\"}], \"markers\": [{\"edgecolor\": \"#000000\", \"facecolor\": \"#FF0000\", \"edgewidth\": 0.5, \"yindex\": 1, \"coordinates\": \"data\", \"zorder\": 2, \"markerpath\": [[[0.0, 5.0], [1.3260155, 5.0], [2.597899353924267, 4.473168457941209], [3.5355339059327378, 3.5355339059327378], [4.473168457941209, 2.597899353924267], [5.0, 1.3260155], [5.0, 0.0], [5.0, -1.3260155], [4.473168457941209, -2.597899353924267], [3.5355339059327378, -3.5355339059327378], [2.597899353924267, 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\"zorder\": 3, \"alpha\": 1, \"fontsize\": 20.0, \"position\": [0.5, 0.90000000000000002], \"rotation\": -0.0, \"id\": \"el827174443024912\"}], \"zoomable\": true, \"images\": [], \"xdomain\": [-4.0, 4.0], \"ylim\": [-0.10000000000000001, 2.1000000000000001], \"paths\": [], \"sharey\": [], \"sharex\": [], \"axesbgalpha\": null, \"axes\": [{\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 9, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 3, \"coordinates\": \"data\", \"dasharray\": \"none\", \"zorder\": 2, \"alpha\": 1, \"xindex\": 0, \"linewidth\": 2, \"data\": \"data01\", \"id\": \"el827174443022672\"}], \"markers\": [{\"edgecolor\": \"#000000\", \"facecolor\": \"#FF0000\", \"edgewidth\": 0.5, \"yindex\": 1, 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" })\n",
" });\n",
"}\n",
"</script>"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x108b9c9d0>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"def piecewise_linear_interp(x,y,xx, fd='centered'):\n",
" \"From samples (x,y) generate piecewise-linear function sampled at points xx using finite difference slopes.\"\n",
" diff = np.abs(x.reshape(1,-1) - xx.reshape(-1,1))\n",
" closest = np.argmin(diff,axis=1)\n",
" \n",
" sigma = np.zeros_like(y)\n",
" if fd == 'centered':\n",
" sigma[1:-1] = (y[2:]-y[:-2])/(x[2:]-x[:-2])\n",
" elif fd == 'forward':\n",
" sigma[:-1] = (y[1:]-y[:-1])/(x[1:]-x[:-1])\n",
" elif fd == 'backward':\n",
" sigma[1:] = (y[1:]-y[:-1])/(x[1:]-x[:-1])\n",
" return y[closest] + sigma[closest]*(xx-x[closest])\n",
"\n",
"def compare_fd(x,y,xx, axis=(-4, 4, -0.1, 2.1)):\n",
" fig, ax = plt.subplots(3,1,figsize=(width,8))\n",
" for i, fd in enumerate( ('centered','forward','backward') ):\n",
" yy = piecewise_linear_interp(x,y,xx,fd=fd)\n",
" ax[i].plot(xx,yy,'-k',lw=2)\n",
" ax[i].hold(True)\n",
" ax[i].plot(x,y,'or',markersize=10,alpha=0.5)\n",
" ax[i].axis( axis );\n",
" ax[i].text(.5,.9,fd,\n",
" horizontalalignment='center',\n",
" transform=ax[i].transAxes,fontsize=20)\n",
" \n",
"compare_fd(x,y,xx)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"I've used $\\sigma=0$ for the points at the edges where we don't have enough data to compute the appropriate slope."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## The problem: overshoots!"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Looking closely, you can see that each of these approximations adds little jumps (called \"overshoots\") in some region where the data itself was monotone. Worse still, each of them generates negative values, whereas the original values were non-negative! If our data represent concentrations or probabilities, then we have no way to make sense of negative values.\n",
"\n",
"Things look even worse if we take data samples from a function that is in fact discontinuous:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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" }(mpld3);\n",
"}else if(typeof define === \"function\" && define.amd){\n",
" // require.js is available: use it to load d3/mpld3\n",
" require.config({paths: {d3: \"https://mpld3.github.io/js/d3.v3.min\"}});\n",
" require([\"d3\"], function(d3){\n",
" window.d3 = d3;\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
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3.909297426825682, 3.909297426825682]]}, \"id\": \"el827174438024784\"});\n",
" });\n",
" });\n",
"}else{\n",
" // require.js not available: dynamically load d3 & mpld3\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/d3.v3.min.js\", function(){\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
" mpld3.draw_figure(\"fig_el827174438024784452180368\", {\"axes\": [{\"xlim\": [-4.0, 4.0], \"yscale\": \"linear\", \"axesbg\": \"none\", \"texts\": [{\"v_baseline\": \"auto\", \"h_anchor\": \"middle\", \"color\": \"#000000\", \"text\": \"centered\", \"coordinates\": \"axes\", \"zorder\": 3, \"alpha\": 1, \"fontsize\": 20.0, \"position\": [0.5, 0.90000000000000002], \"rotation\": -0.0, \"id\": \"el827174454739600\"}], \"zoomable\": true, \"images\": [], \"xdomain\": [-4.0, 4.0], \"ylim\": [-0.5, 4.7999999999999998], \"paths\": [], \"sharey\": [], \"sharex\": [], \"axesbgalpha\": null, \"axes\": [{\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 9, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 1, \"coordinates\": 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false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 9, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 2, \"coordinates\": \"data\", \"dasharray\": \"none\", \"zorder\": 2, \"alpha\": 1, \"xindex\": 0, \"linewidth\": 2, \"data\": \"data01\", \"id\": \"el827174454790992\"}], \"markers\": [{\"edgecolor\": \"#000000\", \"facecolor\": \"#FF0000\", \"edgewidth\": 0.5, \"yindex\": 1, \"coordinates\": \"data\", \"zorder\": 2, \"markerpath\": [[[0.0, 5.0], [1.3260155, 5.0], [2.597899353924267, 4.473168457941209], [3.5355339059327378, 3.5355339059327378], [4.473168457941209, 2.597899353924267], [5.0, 1.3260155], [5.0, 0.0], [5.0, -1.3260155], [4.473168457941209, -2.597899353924267], [3.5355339059327378, -3.5355339059327378], [2.597899353924267, -4.473168457941209], [1.3260155, -5.0], [0.0, -5.0], [-1.3260155, -5.0], [-2.597899353924267, -4.473168457941209], [-3.5355339059327378, -3.5355339059327378], [-4.473168457941209, -2.597899353924267], [-5.0, -1.3260155], [-5.0, 0.0], [-5.0, 1.3260155], [-4.473168457941209, 2.597899353924267], [-3.5355339059327378, 3.5355339059327378], [-2.597899353924267, 4.473168457941209], [-1.3260155, 5.0], [0.0, 5.0]], [\"M\", \"C\", \"C\", \"C\", \"C\", \"C\", \"C\", \"C\", \"C\", \"Z\"]], \"alpha\": 0.5, \"xindex\": 0, \"data\": \"data02\", \"id\": \"el827174454791568pts\"}], \"id\": \"el827174453697168\", \"ydomain\": [-0.5, 4.7999999999999998], \"collections\": [], \"xscale\": \"linear\", \"bbox\": [0.125, 0.39852941176470591, 0.77500000000000002, 0.2279411764705882]}, {\"xlim\": [-4.0, 4.0], \"yscale\": \"linear\", \"axesbg\": \"none\", \"texts\": [{\"v_baseline\": \"auto\", \"h_anchor\": \"middle\", \"color\": \"#000000\", \"text\": \"backward\", \"coordinates\": \"axes\", \"zorder\": 3, \"alpha\": 1, \"fontsize\": 20.0, \"position\": [0.5, 0.90000000000000002], \"rotation\": -0.0, \"id\": \"el827174454887824\"}], \"zoomable\": true, \"images\": [], \"xdomain\": [-4.0, 4.0], \"ylim\": [-0.5, 4.7999999999999998], \"paths\": [], \"sharey\": [], \"sharex\": [], \"axesbgalpha\": null, \"axes\": [{\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 9, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 3, \"coordinates\": \"data\", \"dasharray\": \"none\", \"zorder\": 2, \"alpha\": 1, \"xindex\": 0, \"linewidth\": 2, \"data\": \"data01\", \"id\": \"el827174454840464\"}], \"markers\": [{\"edgecolor\": \"#000000\", \"facecolor\": \"#FF0000\", \"edgewidth\": 0.5, \"yindex\": 1, \"coordinates\": \"data\", \"zorder\": 2, \"markerpath\": [[[0.0, 5.0], [1.3260155, 5.0], 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0.12499999999999989, 0.77500000000000002, 0.22794117647058831]}], \"height\": 640.0, \"width\": 960.0, \"plugins\": [{\"type\": \"reset\"}, {\"enabled\": false, \"button\": true, \"type\": \"zoom\"}, {\"enabled\": false, \"button\": true, \"type\": \"boxzoom\"}], \"data\": {\"data02\": [[-4.0, 0.09070257317431829], [-3.0, 0.0025050133959455545], [-2.0, 0.1585290151921035], [-1.0, 0.520574461395797], [0.0, 1.0], [1.0, 3.479425538604203], [2.0, 3.8414709848078967], [3.0, 3.9974949866040546], [4.0, 3.909297426825682]], \"data01\": [[-4.0, 0.09070257317431829, 0.09070257317431829, 0.09070257317431829], [-3.991991991991992, 0.09070257317431829, 0.08999628640932633, 0.09070257317431829], [-3.983983983983984, 0.09070257317431829, 0.08928999964433433, 0.09070257317431829], [-3.975975975975976, 0.09070257317431829, 0.08858371287934237, 0.09070257317431829], [-3.967967967967968, 0.09070257317431829, 0.08787742611435038, 0.09070257317431829], [-3.95995995995996, 0.09070257317431829, 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3.909297426825682, 3.909297426825682]]}, \"id\": \"el827174438024784\"});\n",
" })\n",
" });\n",
"}\n",
"</script>"
],
"text/plain": [
"<matplotlib.figure.Figure at 0x10886e250>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"y = np.sin(x/2.)+1. + 2.*(x>0)\n",
"compare_fd(x,y,xx,axis=(-4,4,-0.5,4.8))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Now all three approaches have large, obvious overshoots. This becomes even more problematic when actually solving a hyperbolic PDE; see [Lesson 3 of my HyperPython course](http://nbviewer.ipython.org/github/ketch/HyperPython/blob/master/Lesson_03_High-resolution_methods.ipynb) for details.\n",
"\n",
"Is there a better way?"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## The simplest limiter: Minmod"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We'd like to avoid those overshoots and ensure that monotone regions of the data give monotone interpolations. We can do that by choosing the slope $\\sigma_i$ small enough that the interpolant near $x_i$ stays bounded between the neighboring averages $(y_{i-1}+y_i)/2$ and $(y_i+y_{i+1})/2$. There's an easy way to do that: just compute the forward and backward differences (like we did above), and then use *whichever is smaller* in absolute value. If $y_i$ is an extremum, then to avoid increasing the overall range of the data we always choose $\\sigma_i=0$.\n",
"\n",
"Here's what that looks like:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def pw_minmod(x,y,xx):\n",
" \"From samples (x,y) generate piecewise-linear function sampled at points xx using Minmod slopes.\"\n",
" diff = np.abs(x.reshape(1,-1) - xx.reshape(-1,1))\n",
" closest = np.argmin(diff,axis=1)\n",
" \n",
" forward = np.zeros_like(y)\n",
" backward = np.zeros_like(y)\n",
" sigma = np.zeros_like(y)\n",
" \n",
" forward[:-1] = (y[1:]-y[:-1])/(x[1:]-x[:-1])\n",
" backward[1:] = (y[1:]-y[:-1])/(x[1:]-x[:-1])\n",
"\n",
" sigma = (np.sign(forward)+np.sign(backward))/2. * np.minimum(np.abs(forward),np.abs(backward))\n",
" \n",
" return y[closest] + sigma[closest]*(xx-x[closest])"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
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" }(mpld3);\n",
"}else if(typeof define === \"function\" && define.amd){\n",
" // require.js is available: use it to load d3/mpld3\n",
" require.config({paths: {d3: \"https://mpld3.github.io/js/d3.v3.min\"}});\n",
" require([\"d3\"], function(d3){\n",
" window.d3 = d3;\n",
" mpld3_load_lib(\"https://mpld3.github.io/js/mpld3.v0.3git.js\", function(){\n",
" \n",
" mpld3.draw_figure(\"fig_el8271744549417767016057601\", {\"axes\": [{\"xlim\": [-4.0, 4.0], \"yscale\": \"linear\", \"axesbg\": \"none\", \"texts\": [{\"v_baseline\": \"auto\", \"h_anchor\": \"middle\", \"color\": \"#000000\", \"text\": \"Minmod approximation\", \"coordinates\": \"axes\", \"zorder\": 3, \"alpha\": 1, \"fontsize\": 20.0, \"position\": [0.5, 1.0224014336917562], \"rotation\": -0.0, \"id\": \"el827174455981328\"}], \"zoomable\": true, \"images\": [], \"xdomain\": [-4.0, 4.0], \"ylim\": [-0.5, 4.7999999999999998], \"paths\": [], \"sharey\": [], \"sharex\": [], \"axesbgalpha\": null, \"axes\": [{\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"bottom\", \"nticks\": 9, \"tickvalues\": null}, {\"scale\": \"linear\", \"tickformat\": null, \"grid\": {\"gridOn\": false}, \"fontsize\": 22.0, \"position\": \"left\", \"nticks\": 7, \"tickvalues\": null}], \"lines\": [{\"color\": \"#000000\", \"yindex\": 1, 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