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plotting - Mathematica taking forever to compute


I want to simulate the fresnel diffraction through circular aperture.But mathematica is taking forever to calculate.Here is my code.


  f[x_?NumericQ, y_?NumericQ, d_?NumericQ, p_?NumericQ, t_?NumericQ] := 
 NIntegrate[(\[Zeta]*(Exp[
       I*20000 ((d^2 + \[Zeta]^2)^(0.5) + ((p^2 + (x - \[Zeta]*
                  Cos[t])^2 + (y - \[Zeta]*
                  Sin[t])^2)^(0.5)))]))/(d^2 + \[Zeta]^2)^(0.5), {\
\[Zeta], 0, 0.008}]

w[x_?NumericQ, y_?NumericQ, d_?NumericQ, p_?NumericQ] := 
 NIntegrate[f[x, y, d, p, t], {t, 0, 2*Pi}]
g[x_, y_, d_, p_] := (Abs[w[x, y, d, p]])^2
Plot3D[g[x, y, 0.004, 20], {x, -100, 100}, {y, -100, 100}, 
 PlotRange -> Full]

What's the way out? *Specs:i5 7th gen intel processor,12 gb ram



Answer



Let's compile the integrand into a Listable CompiledFunction:


cintegrand = Block[{x, y, d, p, ζ, t},

   With[{code = 
      N[(ζ (Exp[I 20000 ((d^2 + ζ^2)^(1/2) + ((p^2 + (x - ζ Cos[t])^2 + (y - ζ Sin[t])^2)^(1/2)))]))/(d^2 + ζ^2)^(1/2)]
     },
    Compile[{{x, _Real}, {y, _Real}, {d, _Real}, {p, _Real}, {ζ, _Real}, {t, _Real}},
     code,
     CompilationTarget -> "C",
     RuntimeAttributes -> {Listable},
     Parallelization -> True
     ]
    ]

   ];

Next, pick a Gauss quadrature rule for high order polynomials (the integrand is very smooth but quite oscillatory):


{pts, weights, errweights} = NIntegrate`GaussRuleData[9, $MachinePrecision];

Divide the integration domain into $m \times n$ rectangles and set up the quadrature points and weights:


m = 20;
n = 20;
ζdata = Partition[Subdivide[0., 0.008, m], 2, 1];
tdata = Partition[Subdivide[0., 2 Pi, n], 2, 1];

{ζ, t} = Transpose[Tuples[{Flatten[ζdata.{1. - pts, pts}], Flatten[tdata.{1. - pts, pts}]}]];
ω = Flatten[KroneckerProduct[
    Flatten[KroneckerProduct[Differences /@ ζdata, weights]],
    Flatten[KroneckerProduct[Differences /@ tdata, weights]]
    ]];

Define the mapping that parameterizes the graph of OP's function w:


W = {x, y} \[Function] {x, y, Abs[cintegrand[x, y, 0.004, 20., ζ, t].ω]^2};

Compute the values of W on a $101 \times 101$ grid:



R = 2.;
data = Outer[W, Subdivide[-R, R, 100], Subdivide[-R, R, 100]]; // AbsoluteTiming // First


28.2668



Plot the result


ListPlot3D[Flatten[data, 1], PlotRange -> All, AxesLabel -> {"x", "y", "w"}]

enter image description here



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