求大神看看是MEID:AB0A8PESN:感觉被骗了【华为吧】_百度贴吧
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求大神看看是MEID:A000收藏
求大神看看是MEID:AB0A8PESN:感觉被骗了
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Cell[BoxData[
\(Ya\ est\[AAcute]\ hecho\ el\ cambio\ \((uno\ chido\ es\ tfinal\
4)\)\)], "Input"],
Cell[TextData[{
"We start by evaluating these basic definitions.
The number of spatial \
grid points (here, 500), the range of ",
StyleBox["x",
FontSlant->"Italic"],
"-values (here, -10 to 10), the lattice spacing ",
StyleBox["k",
FontSlant->"Italic"],
", and the grid of ",
StyleBox["x",
FontSlant->"Italic"],
"-values."
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FontColor->RGBColor[0, 0, 1]],
Cell[BoxData[
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Cell[BoxData[
\(\(xm = \(-30\);\)\)], "Input"],
Cell[BoxData[
\(\(xp = 30;\)\)], "Input"],
Cell[BoxData[
\(\(kval = \((xp - xm)\)/\)\)], "Input"],
Cell[BoxData[
\(\(xs = Table[xm + kval*j, {j, 1, pts}];\)\)], "Input"],
Cell["Load the package for plotting multiple point sets.", "Text",
FontColor->RGBColor[0, 0, 1]],
Cell[BoxData[
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Cell[BoxData[
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Cell[TextData[{
"We will now consider numerical integration of the Korteweg-de Vries (KdV) \
equation",
StyleBox[" ", "InlineFormula"],
StyleBox[Cell[BoxData[
\(TraditionalForm\`u\_t\)], "InlineFormula",
FontSlant->"Italic"], "InlineFormula"],
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Cell[CellGroupData[{
Cell["Experiment for the Lax-Levermore article", "Title",
FontSize->24,
FontColor->RGBColor[0, 0, 1]],
Cell[TextData[{
"The following cell defines one time-step forward of size ",
StyleBox["h",
FontSlant->"Italic"],
" using the Euler method.
To solve the partial differential equation the \
StyleBox[" ",
FontSlant->"Italic"],
Cell[BoxData[
\(TraditionalForm\`u\_x\)],
FontSlant->"Italic"],
Cell[BoxData[
StyleBox[\(u\_xxx\),
FontSlant->"Italic"], TraditionalForm]]],
"are discretized by symmetric differences with grid spacing ",
StyleBox["k",
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The input and output have the same form, a vector ",
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FontSlant->"Italic"],
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\(\(KdVEulerstep[u_, h_, k_, epsilon_] :=
u + 6*\((h/\((2*k)\))\)*\((\((RotateLeft[u] + u + RotateRight[u])\)/
3)\)*\((RotateLeft[u] - RotateRight[u])\) -
epsilon^2*\((h/\((2*k^3)\))\)*\((RotateLeft[u, 2] -
2*RotateLeft[u] + 2*RotateRight[u] -
RotateRight[u, 2])\);\)\)], "Input"],
Cell[TextData[{
"As in our integration of the equations resembling the wave equation, after \
taking one time step forward by the Euler method, we will take all remaining \
steps using the leapfrog method.
To implement one step of the leapfrog \
method we need to know not only the current values of ",
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", but also their values one time-step ago.
Hence the component arrays ",
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" of the input and output structure.
The next cell contains the
definition of one time-step forward of size ",
StyleBox["h",
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" using the leapfrog method."
}], "Text",
FontColor->RGBColor[0, 0, 1]],
Cell[BoxData[
\(\(KdVLeapfrogstep[unewold_, h_, k_,
epsilon_] := \((uold = unewold[\([2]\)];
unew = unewold[\([1]\)]; {uold +
6*\((2*h/\((2*k)\))\)*\((\((RotateLeft[unew] + unew +
RotateRight[unew])\)/3)\)*\((RotateLeft[unew] -
RotateRight[unew])\) -
epsilon^2*\((2*h/\((2*k^3)\))\)*\((RotateLeft[unew, 2] -
2*RotateLeft[unew] + 2*RotateRight[unew] -
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"Evolving the solution of the wave equation forward for a finite time ",
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The following cell contains defines the result of such \
an evolution."
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\(\(KdVevolve[u_, t_, tpts_, k_, epsilon_] := \((ic = h = t/
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soln[\([1]\)])\);\)\)], "Input"],
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"As an initial condition, let us pick ",
StyleBox["u(x,0)=1+",
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StyleBox["cos",
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StyleBox["(2\[Pi]x/20).",
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You can change this part to replace ",
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}], "Text",
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Cell[TextData[{
"The variable ",
StyleBox["ulater", "Input",
FontColor->GrayLevel[0]],
" will contain the numerical approximation to the solution at various times \
in the future.
We initialize it to the initial condition here."
}], "Text",
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\(\(ulater =\)\)], "Input"],
Cell[TextData[{
"Having evaluated all of the cells above we are ready to run our numerical \
scheme and view the results.
The following cells contain define a value for \
the parameter \[Epsilon] and a time step for which the numerical integration \
is stable and auxiliary quantities needed for the numerical integration.
StyleBox["Do", "Input",
FontColor->GrayLevel[0]],
" loop then repeatedly plots the state of the system as a red curve for ",
StyleBox["u(x,t)",
FontSlant->"Italic"],
as a function of ",
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" (on the left) and the square modulus of the Fourier coefficients of the \
solution as a function of mode number (on the right) and then advances the \
solution numerically to a time ",
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" units in the future.
This produces a sequence of frames of a movie \
showing the evolution of ",
StyleBox["u",
FontSlant->"Italic"],
" and its Fourier spectrum under the KdV equation.
The feature to notice \
here is that when \[Epsilon] is going to zero, the solution at some fixed \
time tfinal, tends to some function almost everywhere, fot t enough large. \
Also, notice that ..."
}], "Text",
FontColor->RGBColor[0, 0, 1]],
Cell[TextData[{
"In the Lax-Levermore article they analize the KdV equation\n
StyleBox[Cell[BoxData[
\(TraditionalForm\`u\_t\)], "InlineFormula",
FontSlant->"Italic"], "InlineFormula"],
StyleBox["-6", "InlineFormula"],
StyleBox["u", "InlineFormula",
FontSlant->"Italic"],
Cell[BoxData[
\(TraditionalForm\`u\_\(\(x\)\(\ \)\) + \[Epsilon]\^2\)]],
StyleBox[Cell[BoxData[
\(TraditionalForm\`u\_xxx\)], "InlineFormula",
FontSlant->"Italic"], "InlineFormula"],
StyleBox["=0,\n\nwith a initial condition u(x,0)=u(x) and \
\[Epsilon]\[Rule]0. We will do it numerically and we will try to interpret \
the results and see that they are agree with the results in the artcile. In \
the data below, we start with an initial epsilon called \"epsilonval\", then \
we find with the method explained above the solution of the equation with \
this initial epsilon at fixed time \"tfinal\". Then we change the value of \
epsilon for many epsilon and epsilon tends to zero, we stop at the value \
\"EpsilonFinal\" and this should be small enough. \"NumberEpsilonSteps\" is \
the number of different values for epsilon from epsilonval to EpsilonFinal. \
The increment is constant, this increment is given by \"Elen\". \"EpsilonStep\
\" will be the different values of epsilon form epsilonval through \
EpsilonFinal and we start with EpsilonStep=epsilonval. \"hval\" is the val of \
h for the Euler Method. \"tlen\" is the increment of t for arrive from t=0 \
(in the initial condition) to tfinal. I don\.b4t know what \"steps\" is, but \
it is explained in the Euler method. We already define what \"tfinal\" is. \
Finally, \"NumberSteps\" give us how many steps we need (one graph on each \
step) to arrive to final from t=0.", "InlineFormula",
FontSlant->"Italic"]
}], "Text",
FontColor->RGBColor[0, 0, 1]],
Cell[BoxData[{
\(\(epsilonval = 1;\)\), "\[IndentingNewLine]",
\(\(NumberEpsilonSteps = 25;\)\), "\[IndentingNewLine]",
\(\(EpsilonFinal = 0.01;\)\), "\[IndentingNewLine]",
\(\(Elen = \(epsilonval - EpsilonFinal\)\/NumberEpsilonS\)\), "\
\[IndentingNewLine]",
\(\(EpsilonStep =\)\)}], "Input"],
Cell[BoxData[
\(\(hval = 0.1*kval^3/epsilonval^2;\)\)], "Input"],
Cell[BoxData[
\(\(tlen = 0.06;\)\)], "Input"],
Cell[BoxData[
\(\(steps = Round[tlen/hval];\)\)], "Input"],
Cell[CellGroupData[{
Cell[BoxData[{
\(\(tfinal = 1.7;\)\), "\[IndentingNewLine]",
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Cell[BoxData[
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Cell[TextData[{
In the Lax-Levermore article, some results is that before a critical \
time t"Italic"],
") tend to a solution of the equation for ",
StyleBox["\[Epsilon] ", "InlineFormula",
FontSlant->"Italic"],
"= 0, and that that convergence is L^2 strong. Another result is that for t \
large enough, u(x,t;",
StyleBox["\[Epsilon]", "InlineFormula",
FontSlant->"Italic"],
") tend to a function u(x,t) but in a L^2 weak sense. So, we can change the \
values of \"tfinal\" above (enough small and enough large) and see the \
difference. What we do below is plot the initial condition, plot the \
different solutions of the KdV equation at a fixed time tfinal and for the \
different values of epsilon. Finally, we find numerically the solution of the \
equation but for ",
StyleBox["\[Epsilon]", "InlineFormula",
FontSlant->"Italic"],
}], "Text",
FontColor->RGBColor[0, 0, 1]],
Cell[CellGroupData[{
Cell[BoxData[{
\(Plot[\(-\((3/2)\)\)*Sech[\((1/12)\)*x]^2, {x, \(-30\),
30}]\), "\[IndentingNewLine]",
\(Do[For[i = 1, i < NumberSteps + 1,
KdVevolve[ulater, tlen, steps, kval,
EpsilonStep]; \(i++\)]; \[IndentingNewLine]direct =
ListPlot[Transpose[{xs, ulater}], PlotJoined \[Rule] True,
PlotRange \[Rule] {{\(-30\), 30}, {\(-2\), 1}},
PlotStyle \[Rule] Hue[0],
DisplayFunction \[Rule] Identity]; \[IndentingNewLine]fourier =
MultipleListPlot[RotateRight[Abs[Fourier[ulater]]^2, pts/2],
Axes \[Rule] False, SymbolShape \[Rule] Stem,
SymbolStyle \[Rule] Hue[0],
PlotRange \[Rule] {{225, 275}, {\(-10\), 510}},
DisplayFunction \[Rule] Identity];
Show[GraphicsArray[{direct, fourier}],
DisplayFunction \[Rule] $DisplayFunction, ImageSize \[Rule] 2*288];
EpsilonStep = EpsilonStep - E
ulater = uinit, {NumberEpsilonSteps}]\), "\[IndentingNewLine]",
\(\(ulater =\)\), "\[IndentingNewLine]",
\(\(For[i = 1, i < NumberSteps + 1,
KdVevolve[ulater, tlen, steps, kval,
0]; \(i++\)];\)\), "\[IndentingNewLine]",
\(\(direct =
ListPlot[Transpose[{xs, ulater}], PlotJoined \[Rule] True,
PlotRange \[Rule] {{\(-30\), 30}, {\(-2\), 1}},
PlotStyle \[Rule] Hue[0],
DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]",
\(fourier =
MultipleListPlot[RotateRight[Abs[Fourier[ulater]]^2, pts/2],
Axes \[Rule] False, SymbolShape \[Rule] Stem,
SymbolStyle \[Rule] Hue[0],
PlotRange \[Rule] {{225, 275}, {\(-10\), 510}},
DisplayFunction \[Rule] Identity];
Show[GraphicsArray[{direct, fourier}],
DisplayFunction \[Rule] $DisplayFunction,
ImageSize \[Rule] 2*288];\[IndentingNewLine]\)}], "Input"],
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