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"source": [
"Finite Differences Lab Example"
]
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"source": [
"Aron Ahmadia (US Army ERDC) and David Ketcheson (KAUST)"
]
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{
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"level": 3,
"metadata": {},
"source": [
"Teaching Numerical Methods with IPython Notebooks, SciPy 2014"
]
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"source": [
" This example by Aron Ahmadia and David Ketcheson is licensed under a Creative Commons Attribution 4.0 International License. All code examples are also licensed under the [MIT license](http://opensource.org/licenses/MIT)."
]
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"from IPython.core.display import HTML\n",
"css_file = './example.css'\n",
"HTML(open(css_file, \"r\").read())"
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"html": [
"\n",
"\n",
"\n",
"\n",
"\n",
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""
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"Together with your teammate, write a Python script or function to do the following:\n",
"\n",
"1) Generate a set of $m+2$ equally-spaced points $0=x_0,x_1,x_2,\\dots,x_m+1=1$. \n",
"2) Evaluate the following function at those points: \n",
"\n",
"$$f(x) = 1 + x + 3x^2 + 2\\sin(5x)$$\n",
"\n",
"3) Evaluate the finite difference approximation\n",
"\n",
"$$f''(x) \\approx \\frac{f(x_{i-1}) - 2 f(x_i) + f(x_{i+1})}{h^2}$$\n",
"\n",
"for $i=2,3,\\dots,m$, where $h$ is the spacing between successive points.\n",
"\n",
"4) Plot the finite difference approximation and the exact values of the second derivative.\n",
"\n",
"Try running your function with different values of $m$ and check that its output tends to the exact second derivative as $m$ is increased.\n",
"\n",
"There are at least 3 different ways to program step 3:\n",
"\n",
"- Using a loop\n",
"- Using matrix-vector multiplication\n",
"- Using array slicing\n",
"\n",
"Can you code all 3? Which is the fastest? You can time any line of code in IPython using %timeit."
]
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"Now add the following to your function:\n",
"\n",
"1) Compute the error of your approximation in the following grid-function norms:\n",
"\n",
"- $L_1: h\\sum_i |E(x_i)|$\n",
"- $L_2: \\sqrt{h\\sum_i \\left(E(x_i)\\right)^2}$\n",
"- $L_\\infty$: $\\max_i |E(x_i)|$\n",
"\n",
"2) Loop over the following values of $m$: $1,2,4,8,16$. \n",
"Compute the three error norms for each $m$-value.\n",
"\n",
"3) Estimate the convergence rate using the formula $E(h) \\approx Ch^p$. *Hint: take the absolute value of both sides and then take the log. What does the resulting equation describe, and what is the role of $p$?*"
]
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