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plotting - Testing the accuracy of numerically computed derivatives


I am calculating approximate derivatives by using NDSolve`FiniteDifferenceDerivative, so this works:


 Subscript[Der, i_][yyy_] := 
   Module[{xx}, 
     xx = Length[yyy];
     NDSolve`FiniteDifferenceDerivative[

       Derivative[i], 
       N[yyy], 
       DifferenceOrder -> 2] @ "DifferentiationMatrix"
         // Normal // Developer`ToPackedArray // SparseArray];

xi = 1.;
xf = -1;
yy = 100; 
xgrid = Table[xi + i (xf - xi/yy), {i, 0, yy}];


(Der1 = Subscript[Der, 1][xgrid]) // MatrixForm;

numerical = Der1.Exp[-xgrid^2];
exact = -2*xgrid*Exp[-xgrid^2];

diff = numerical - exact; 
diffError = yy^2*diff

ListLinePlot[yy^2 Abs[diff]]


I want to show my solution is accurate by demonstrating that the difference between the numerical solution and the exact solution goes to zero as $\mathtt{yy}^{-2}$. For this I want to plot $\mathtt{yy}^2 |\mathrm{numerical} - \mathrm{exact}|$ for different values of $\mathtt{yy}$ but am not sure how to do this.


The code gives reasonable values for the differences, though I am not sure how to plot them for different $\mathtt{yy}$ values.


I obtained the follow plot from the code shown above.


accuracytest



Answer



xi = -1.;
xf = 1;
xgrid[yy_] := Range[xi, xf, (xf - xi)/yy]

Subscript[Der, i_][yyy_] := Module[{xx}, xx = Length[yyy];

 NDSolve`FiniteDifferenceDerivative[Derivative[i], N[yyy], 
    DifferenceOrder -> 2]@"DifferentiationMatrix" // Normal // 
 Developer`ToPackedArray // SparseArray];

Der1[yy_] := Subscript[Der, 1][xgrid[yy]]
numerical[yy_] := Der1[yy].Exp[-xgrid[yy]^2]
exact[yy_] := -2*xgrid[yy]*Exp[-xgrid[yy]^2]
diff[yy_] := numerical[yy] - exact[yy]

yyvals = {100, 300, 1000};


ListLinePlot[
 Table[Transpose[{xgrid[yy], yy^2 Abs[diff[yy]]}], {yy, yyvals}], 
 PlotRange -> All, PlotLegends -> StringTemplate["yy = ``"] /@ yyvals]

enter image description here


Max[diff[100]] / Max[diff[1000]] = 99.9756

This means the error ~ 1/yy^2. For better see this scaling low one can use logarithmic scale:


ListLinePlot[

 Table[Transpose[{xgrid[yy], Abs[diff[yy]]}], {yy, yyvals}], 
 PlotRange -> All, PlotLegends -> StringTemplate["yy = ``"] /@ yyvals,
 ScalingFunctions -> "Log", Frame -> True]

enter image description here


NonlinearModelFit[Table[{yy, Max[diff[yy]]}, {yy, 100, 10000, 500}], 
 a + b/x^2, {a, b}, x]


Blockquote




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