(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 78427, 2019]*) (*NotebookOutlinePosition[ 79074, 2041]*) (* CellTagsIndexPosition[ 79030, 2037]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[StyleBox["Problem 6:", "Title"]], "Text"], Cell["\<\ Given an array A which represents a field f(x) return an array B which \ represents f ' (x) using finite difference method. Note we're assuming the \ function is periodic in the interval [0,2 Pi).\ \>", "Text"], Cell[BoxData[ \(\(finitediff[A_] := Module[{B}, \[IndentingNewLine]B = Table[0, {i, 1, n}]; \[IndentingNewLine]For[j = 0, j < n, \(j++\), \ \[IndentingNewLine]B[\([j + 1]\)] = 1/2 \((A[\([Mod[j + 1, n] + 1]\)] - A[\([Mod[j - 1, n] + 1]\)])\)/\((2 Pi/n)\)]; \[IndentingNewLine]Return[ B];\[IndentingNewLine]];\)\)], "Input"], Cell["\<\ We'd like to solve the equation v du/dt+du/dx=0. For simplicity let's set \ v=1. Given an array A, evolve over one time step using Runge-Kutta with \ finite difference method.\ \>", "Text"], Cell[BoxData[ \(runge[A_, dt_] := Module[{k1, k2, k3, k4, B}, \[IndentingNewLine]k1 = \(-dt\)* finitediff[A]; \[IndentingNewLine]k2 = \(-dt\)* finitediff[A + k1/2]; \[IndentingNewLine]k3 = \(-dt\)* finitediff[A + k2/2]; \[IndentingNewLine]k4 = \(-dt\)* finitediff[A + k3]; \[IndentingNewLine]B = A + 1/6 \((k1 + 2 \((k2 + k3)\) + k4)\); \[IndentingNewLine]Return[ B];\[IndentingNewLine]]\)], "Input"], Cell["\<\ The spatial grid points are located at 2 pi j/n, where j is an integer and n \ is the number of grid points. Let's evolve for a total time T.\ \>", "Text"], Cell[BoxData[{ \(\(T = 10;\)\), "\[IndentingNewLine]", \(\(n = 10;\)\)}], "Input"], Cell["\<\ The initial field confguartion will be called u0 and the final field to be \ u1. We'll take it to be u0=exp(cos(x)). Then the exact solution is \ u(x,t)=exp(cos(x-t)) and u1=exp(cos(x-T)) . We define the error as the square \ root of the integral of (u_numerical-u_actual)^2 from x= 0 to x=2 Pi at the \ final time T. Compute the error as a function of time step, varying the time \ step from dx/10 to 4dx in multiples of dx/10, where dx=2 Pi / n.\ \>", "Text"], Cell[BoxData[{ \(\(error = Table[0, {i, 1, 40}];\)\), "\[IndentingNewLine]", \(\(u0 = Table[1. Exp[Cos[2 Pi\ \((j - 1)\)/n]], {j, 1, n}];\)\), "\[IndentingNewLine]", \(For[k = 1, k \[LessEqual] 40, \(k++\), \[IndentingNewLine]u = u0; \[IndentingNewLine]dt = 2 Pi\ k/\((10 n)\); \[IndentingNewLine]t = IntegerPart[T/dt]; \[IndentingNewLine]u1 = Table[1. Exp[Cos[2 Pi\ \((j - 1)\)/n - dt*t]], {j, 1, n}]; \[IndentingNewLine]For[i = 0, i \[LessEqual] t, \(i++\), u = runge[u, dt]]; \[IndentingNewLine]error[\([k]\)] = Sqrt[2 Pi/\ n* Sum[\((u[\([j]\)] - u1[\([j]\)])\)^2, {j, 1, n}]];\[IndentingNewLine]]\)}], "Input"], Cell["\<\ Plot the log of error a function of dt for n=10. 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Runge-Kutta with \ pseudospectral method.\ \>", "Text"], Cell[BoxData[ \(runge[A_, dt_] := Module[{k1, k2, k3, k4, B}, \[IndentingNewLine]k1 = \(-dt\)* pseudospec[A]; \[IndentingNewLine]k2 = \(-dt\)* pseudospec[A + k1/2]; \[IndentingNewLine]k3 = \(-dt\)* pseudospec[A + k2/2]; \[IndentingNewLine]k4 = \(-dt\)* pseudospec[A + k3]; \[IndentingNewLine]B = A + 1/6 \((k1 + 2 \((k2 + k3)\) + k4)\); \[IndentingNewLine]Return[ B];\[IndentingNewLine]]\)], "Input"], Cell["\<\ Compute the error as a function of time step, varying the time step from \ dx/10 to 2dx in multiples of dx/10, where dx=2 Pi / n. 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