(************** 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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Setting up, I have:", FontSize->14]], "Text"], Cell[BoxData[{ FormBox[ RowBox[{\(U\_t\), "=", FormBox[\(U\_xx\), "TraditionalForm"]}], "TraditionalForm"], "\[IndentingNewLine]", FormBox[ RowBox[{\(x\_\(i + 1\)\), "=", FormBox[\(Ax\_i\), "TraditionalForm"]}], "TraditionalForm"], "\[IndentingNewLine]", RowBox[{ FormBox[ RowBox[{\(x\_\(i + 1\)\), " ", "=", FormBox[\(x\_i\), "TraditionalForm"]}], "TraditionalForm"], " ", "+", " ", \(dt\ A\ x\_\(i + 1\)\)}], "\[IndentingNewLine]", RowBox[{\((I\ - \ dt\ A)\), FormBox[ RowBox[{\(x\_\(i + 1\)\), " ", "=", FormBox[\(x\_i\), "TraditionalForm"]}], "TraditionalForm"], " "}]}], "Text"], Cell[TextData[{ "\n", StyleBox["Using routines from the last project, I define the coefficient \ matrix I - dt A below as A:", FontSize->14] }], "Text"], Cell[BoxData[{ \(\(Clear[GaussSeidlStep, \ RunAlgorithm, \ InfinityNorm, \ MultiplyA];\)\), "\[IndentingNewLine]", \(\(GaussSeidlStep[x_, \ b_, \ n_, \ \[Alpha]_]\ := \ \ Block[{u, \ outu, \ i}\ , u\ = \ Table[\(x[\([i]\)]\)[\([2]\)], \ {i, \ 1, \ n}]; For[i = 1, i \[LessEqual] n, \ \(i++\), \ u[\([i]\)]\ = \ \((\(b[\([i]\)]\)[\([2]\)]\ - \ \[Sum]\+\(j = 1\ \)\%n If[Not[j \[Equal] i], A[i, \ j, \ n, \ \[Alpha]]\ u[\([j]\)], 0])\)/ A[i, \ i, \ n, \ \[Alpha]]]; \ outu\ = \ Table[{\(x[\([i]\)]\)[\([1]\)], \ u[\([i]\)]}, \ {i, \ 1, \ n}]; \ outu];\)\), "\[IndentingNewLine]", \(\(RunAlgorithm[algorithm_, \ steps_, \ x_, \ b_, \ n_, \ \[Alpha]_]\ := \ Block[{\ a, \ holdb\ = \ b, \ curr\ = \ x, \ residualnorms, \ errnorms}, \ \ For[a = 1, \ a \[LessEqual] \ steps, \ \(a++\), \ \ curr\ = \ N[algorithm[curr, \ holdb, \ n, \ \[Alpha]]]]; \ curr];\)\), "\[IndentingNewLine]", \(\(InfinityNorm[x_, \ n_]\ := \ Block[{i, \ tmax\ = \ Abs[\(x[\([1]\)]\)[\([2]\)]]}, \ For[i = 1, i \[LessEqual] n, \(i++\), tmax\ = \ Max[tmax, \ \(x[\([i]\)]\)[\([2]\)]]]; \ tmax];\)\), "\[IndentingNewLine]", \(\(MultiplyA[x_, \ n_, \ \[Alpha]_]\ := \ Table[\[Sum]\+\(j = 1\)\%n A[i, \ j, \ n, \[Alpha]]\ \(x[\([j]\)]\)[\([2]\)], \ {i, \ 1, \ n}];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Clear[A];\)\), "\[IndentingNewLine]", \(\(A[i_, \ j_, \ n_, \ h_]\ := \ \ If[ Or[i \[Equal] 1, \ i \[Equal] n], \ If[i \[Equal] j, \ \ 1\ - \(\(\ \)\(h\)\)\/\((2\/n)\)\^2, \ \ If[ Or[j \[Equal] i + 1, j \[Equal] i\ - \ 1], \(\(\ \ \)\(2\ h\)\)\/\((2\/n)\)\^2, \ \ If[ Or[j \[Equal] i\ + \ 2, j \[Equal] i\ - \ 2], \(-\(\(\(\ \)\(h\)\)\/\((2\/n)\)\^2\)\), 0]]], \ \ If[ i \[Equal] j, \ \ 1\ + \(\(\ \)\(2\ h\)\)\/\((2\/n)\)\^2, \ \ If[ Or[j \[Equal] i + 1, j \[Equal] i\ - \ 1], \(-\ \(h\/\((2\/n)\)\^2\)\), 0]]];\)\[IndentingNewLine]\), "\[IndentingNewLine]", \(\(Clear[BackwardEuler];\)\), "\[IndentingNewLine]", \(\(BackwardEuler[X_, \ n_\ , \ dt_, \ iterations_, \ approxits_]\ := Block[{Xcurr\ = X, \ i}, \ For[i\ = \ 0, \ i\ < \ iterations, \ \(i++\), \ Xcurr = \ RunAlgorithm[GaussSeidlStep, \ approxits, \ Xcurr, \ Xcurr, \ n + 1, \ dt]]; \ Xcurr];\)\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ StyleBox["Not surprisingly, I see that the solution rapidly converges to an \ exponential, as f(x)=", FontSize->14], Cell[BoxData[ \(TraditionalForm\`A\ \(\[ExponentialE]\^x\) \[ExponentialE]\^t + B\ \(\[ExponentialE]\^\(-x\)\) \[ExponentialE]\^t\)], FontSize->14], StyleBox[" are solutions. (Below, the simulation has been run for 1 \ time-unit); Apparently, more rapidly-changing functions converge more \ quickly to the exponentials.", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(k = 1;\)\), "\[IndentingNewLine]", \(\(timestep\ = 0.1;\)\), "\[IndentingNewLine]", \(\(numpoints = \ 100;\)\), "\[IndentingNewLine]", \(\(timeits\ = \ 10;\)\), "\[IndentingNewLine]", \(\(mpits\ = \ 10;\)\), "\[IndentingNewLine]", \(\(Xstart\ = \ Table[{N[\((2 i\/numpoints - 1)\)], N[Sin[k\ \((2 i\/numpoints - 1)\)]]}, \ {i, 0, \ numpoints}];\)\), "\[IndentingNewLine]", \(\(Trial2\ = \ BackwardEuler[Xstart, \ numpoints, \ timestep, timeits, \ mpits];\)\), "\[IndentingNewLine]", \(\(ListPlot[ Xstart, \ {PlotJoined \[Rule] True}];\)\), "\[IndentingNewLine]", \(\(ListPlot[ Trial2, \ {PlotJoined \[Rule] True}];\)\), "\[IndentingNewLine]", \(\)}], "Input"], Cell[CellGroupData[{ Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart 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Define \ the algorithm as follows:", FontSize->14] }], "Text"], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(Clear[f, \ ForwardEuler, \ numpoints, \ x, \ i, \ n, \ dt, \ X]\[IndentingNewLine] \(f[x_, \ i_, n_]\ := \ If[i \[Equal] 1, coeffA\ \(\(\(x[\([i + 1]\)]\)[\([2]\)]\)\(\ \)\(-\)\(\ \ \ \)\(\(x[\([i]\)]\)[\([2]\)]\)\(\ \)\)\/\((2\/n)\)\ \ , coeffA\ \(\(x[\([i\ ]\)]\)[\([2]\)]\ - \ \(x[\([i - \ 1]\)]\)[\([2]\)]\)\/\((2\/n)\)];\)\[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[{ "\n", StyleBox["The result seems to be simply a travelling wave for sinusoidal \ initial conditions:", FontSize->14] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(k\ = 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Under the same \ substitutions as last time, I get:\n", FontSize->14], "\n", Cell[BoxData[ RowBox[{ RowBox[{ FormBox[\(U\_t\), "TraditionalForm"], "+", RowBox[{"a", FormBox[\(U\_x\), "TraditionalForm"]}]}], "=", "0"}]]], "\n \n", 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], "\n\n", Cell[BoxData[{ \(TraditionalForm\`\(1\/k\) \((v\_\(j\ , n + 1\) - v\_\(j\ , n\))\) + \ \(a\/h\) \((v\_\(j\ + 1, n\) - v\_\(j\ , n\))\) = 0\), "\[IndentingNewLine]", \(TraditionalForm\`\(1\/k\) \((\[ExponentialE]\^\(\[Lambda]\ k\) - 1)\) + \ \(a\/h\) \((\[ExponentialE]\^\(\(\ \)\(\[ImaginaryI]\ \ \[Beta]\ h\)\) - 1)\) = 0\), "\[IndentingNewLine]", \(TraditionalForm\`s\ = \ k\/h\), "\[IndentingNewLine]", \(TraditionalForm\`\[ExponentialE]\^\(\[Lambda]\ k\) = 1 + a\ \(s( 1\ - \ \[ExponentialE]\^\(\(\ \)\(\[ImaginaryI]\ \[Beta]\ \ h\)\)\ )\)\)}], FontSize->14], "\n\n", StyleBox["Now I see that for stability in this scheme, I need to take a < 0 \ and further I need ", FontSize->14], Cell[BoxData[ \(TraditionalForm\`\(\(0\)\(<\)\) | \(a\ k\)\/\(2\ h\) | \(\(<\)\(2\)\)\ \)]], ". 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