Testing the accuracy of numerically computed derivatives - alternative method

Question

So I have asked this question earlier today: Testing the accuracy of numerically computed derivatives,

This method works well for DifferenceOrder methods but upon reading https://reference.wolfram.com/language/tutorial/NDSolveMethodOfLines.html, I believe it would be more accurate to use DifferenceOrder -> "Pseudospectral" with an appropriate grid.

Implementing the code as posted in my question but with the following changes:

Subscript[Der, i_][yyy_] := 
  Module[{xx}, 
    xx = Length[yyy];
    NDSolve`FiniteDifferenceDerivative[
      Derivative[i], 
      N[yyy], 
      DifferenceOrder -> "Pseudospectral"] @ "DifferentiationMatrix"
         // Normal // Developer`ToPackedArray // SparseArray];

xi = 1.;
xf = -1;
yy = 100; 
xgrid = Table[(a+b)/2 + (a-b)/2)*Cos[i \pi/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 = Exp[yy]*diff

I get a very accurate result for the difference though,when I try to plot it, by applying the method prompted in the answer to my previous question, I get an error message:

NDSolve`FiniteDifferenceDerivative::spc: The coordinates {-0.498973, -0.495895,-0.49078, -0.483647, -0.474528, -0.463458, -0.450484, -0.435659, -0.419044, -0.400707, <<29>>, 0.419044, 0.435659, 0.450484, 0.463458, 0.474528, 0.483647, 0.49078, 0.495895, 0.498973, 0.5} for dimension 1 do not have spacing proportional to the Chebyshev points -Cos[Range[0,48] Pi/48]. Nonperiodic pseudospectral derivatives are only computed for this spacing.

It seems to not like my xgrid choice though for when I implement my code the results are really great for the derivative, so it would be great to show this for different yy values. Thanks in advance.

Answer 1

As mentioned in referenced help page you need to construct a special grid:

Subscript[Der, i_][yyy_] := Module[{xx}, xx = Length[yyy];
 NDSolve`FiniteDifferenceDerivative[Derivative[i], N[yyy], 
    DifferenceOrder -> "Pseudospectral"]@"DifferentiationMatrix" //
   Normal // Developer`ToPackedArray // SparseArray];

xi = -1.;
xf = 1;
xgrid[yy_] := xi + (xf - xi)/2 (1 - Cos[π Range[0, yy - 1]/(yy - 1)]);

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];

And now plotting:

yyvals = {100, 300, 1000, 2000};

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