(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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). 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Define the \ algorithm as follows:\ \>", "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Clear[f, \ ForwardEuler, \ numpoints]\[IndentingNewLine] \(f[x_, \ i_, n_]\ := \ If[i \[Equal] 1, \(\(x[\([i]\)]\)[\([2]\)]\ - \ 2\ \(x[\([i + 1]\)]\)[\([2]\)]\ \ + \ \(x[\([i + 2]\)]\)[\([2]\)]\)\/\((2\/n)\)\^2\ , If[i \[Equal] n + 1, \(\(x[\([i]\)]\)[\([2]\)] - \ 2\ \(x[\([i - \ \ 1]\)]\)[\([2]\)]\ + \ \(x[\([i\ - 2]\)]\)[\([2]\)]\)\/\((2\/n)\)\^2\ , \ \(\(x[\([i\ - \ 1]\)]\)[\([2]\)]\ + \ \(x[\([i + 1]\)]\)[\([2]\)]\ - \ 2\ \ \(x[\([i]\)]\)[\([2]\)]\)\/\((2\/n)\)\^2]];\)\[IndentingNewLine] \(ForwardEuler[X_, \ n_\ , \ dt_, \ iterations_]\ := Block[{Xnext\ \ , Xcurr\ = X, \ i, \ j}, Xnext\ = \ Xcurr; \ For[i\ = \ 0, \ i\ < \ iterations, \ \(i++\), Xnext\ = \ Table[{\(Xcurr[\([j + 1]\)]\)[\([1]\)], \ \(Xcurr[\([j + 1]\)]\)[\([2]\)]\ + \ dt\ f[Xcurr, \ j + 1, \ n]}, \ {j, \ 0, \ numpoints}]; \ Xcurr\ = \ Xnext]; \ Xnext];\)\)\)\)], "Input"], Cell[TextData[{ StyleBox["\nFirst let me consider the stability of the heat-conduction \ equation (as above):\n\nLet ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`U\_t\)], FontSize->14], StyleBox["=", FontSize->14], Cell[BoxData[ \(TraditionalForm\`U\_xx\)], FontSize->14], StyleBox[" (let t = j k for j \[GreaterEqual] 0 and x = i h for 0 \ \[LessEqual] i \[LessEqual] n + 1).\n\nIf I write as trial Fourier solutions \ ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`v\_\(j\ n\) = \(\[ExponentialE]\^\(\[ImaginaryI]\ j\ \ \ \[Beta]\ h\)\) \[ExponentialE]\^\(n\ \[Lambda]\ k\)\)], FontSize->14], StyleBox[" with ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\[ImaginaryI] = \@1\)], FontSize->14], StyleBox[" , then with Simpson's rule I get:\n\n", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(1\/h\^2\) \((v\_\(i + 1, \ j\) - 2\ v\_\(i, j\) + v\_\(i - 1, \ j\))\) = \(1\/k\) \((v\_\(i, \ j + 1\) - v\_\(i, \ j\))\)\)], FontSize->14], StyleBox["\n\nwith ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`s = k\/h\^2\)], FontSize->14], StyleBox[", I have:\n\n", FontSize->14], Cell[BoxData[ \(TraditionalForm\`v\_\(i, \ j + 1\) = s\ v\_\(i - 1, \ j\)\ + \ \((1\ - \ 2\ s)\) v\_\(i, \ j\)\ + \ s\ v\_\(i + 1, \ j\)\)], FontSize->14], StyleBox["\n\nSubstituting ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`v\_\(j\ n\) = \(\[ExponentialE]\^\(\[ImaginaryI]\ j\ \ \[Beta]\ h\)\) \[ExponentialE]\^\(n\ \[Lambda]\ k\)\)], FontSize->14], StyleBox[" and simplifying, now I have:\n\n", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\[ExponentialE]\^\(\[Lambda]\ k\) = \(s\ \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\(\ \)\(\[Beta]\)\(\ \)\(h\)\(\ \)\)\ + 1\ - \ 2\ s\ + \ s\ \[ExponentialE]\^\(\(\[ImaginaryI]\)\(\ \)\(\[Beta]\)\(\ \)\(h\ \)\(\ \)\) = 1 - 4\ s\ \(\(sin\^2\)(\(\[Beta]\ h\)\/2)\)\)\)], FontSize->14], StyleBox["\n\nConstraining ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(|\)\(\[ExponentialE]\^\(\[Lambda]\ \ k\)\)\(|\)\(\(\[LessEqual]\)\(1\)\)\)\)], FontSize->14], StyleBox[", then, I see that I must have ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(s\ \(\(sin\^2\)(\(\[Beta]\ h\)\/2)\) \ \[LessEqual] 1\/2\)\)\)], FontSize->14], StyleBox[" so that ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(\ \)\(s \[LessEqual] 1\/2\)\)\)], FontSize->14], StyleBox[" for stability.", FontSize->14], StyleBox["\n\nThis first test will have s = ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`0.1\/0.02\^2\)]], " ", StyleBox["and should be unstable:", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(k\ = 9;\)\[IndentingNewLine] \(timestep\ = 0.1;\)\[IndentingNewLine] \(numpoints = \ 100;\)\[IndentingNewLine] \(totaltimesteps\ = \ 10;\)\[IndentingNewLine] \(Xstart\ = \ Table[{N[\((2 i\/numpoints - 1)\)], N[Sin[k\ \((2 i\/numpoints - 1)\)]]}, \ {i, 0, \ numpoints}];\)\[IndentingNewLine] \(Trial1\ = \ ForwardEuler[Xstart, \ numpoints, \ timestep, totaltimesteps];\)\[IndentingNewLine] \(ListPlot[Trial1, \ {PlotJoined \[Rule] 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Note the \ dramatic difference between this and the result obtained for the same case in \ Backward Euler. Due to the much longer runtime required by the very small \ time-step, I will focus more heavily on the always-converging Backward \ Euler.", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(\(k\ = \ 9;\)\[IndentingNewLine] \(timestep\ = 0.0001;\)\[IndentingNewLine] \(numpoints = \ 100;\)\[IndentingNewLine] \(totaltimesteps\ = \ 10000;\)\[IndentingNewLine] \(Xstart\ = \ Table[{N[\((2 i\/numpoints - 1)\)], N[Sin[k\ \((2 i\/numpoints - 1)\)]]}, \ {i, 0, \ numpoints}];\)\[IndentingNewLine] \(Trial1\ = \ ForwardEuler[Xstart, \ numpoints, \ timestep, totaltimesteps];\)\[IndentingNewLine] \(ListPlot[Trial1, \ {PlotJoined \[Rule] True}];\)\)\)\)], "Input"], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier 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