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How to increase performance of this code for plotting a contour plot?


I have an equation which I need to triple integrate over a unit cube. The equation is


pot = NIntegrate[1/Sqrt[(x - h)^2 + (y - k)^2 + (z - l)^2], {h,-1,1}, {k,-1, 
1},{l,-1,1}];

As soon as I enter Shift+Enter it immediately processes the command. But now what I want is to plot its ContourPlot for different ${z}$ values (I chose $z=0.5$). So I give the command


ContourPlot[pot /. {z -> 0.5}, {x, -2, 2}, {y, -2, 2}]


But this piece of code takes just forever to process. I just keep on waiting and waiting but processing never ends (it takes really really long time). I am not sure that how is this such a computationally heavy task. For $z$ other than $0$ it takes longer time.


Is there something that I am doing wrong? I don't think this is a drawback of the device I am using. Is there a way to improve the performance of this code I am using?


P.S. It's been more than 10 minutes but the code for $z=0.5$ has not processed.


For your reference, I am attaching the contour plot for $z=0$.


enter image description here


This is the output for $z=0.5$ from the code above (it took about 10 minutes)


enter image description here



Answer



One helpfull rule to get fast integration is, to do analytical integration as much as you can.


int1 = Integrate[1/Sqrt[(x - h)^2 + (y - k)^2 + (z - l)^2], {l, -1, 1}, 

         Assumptions -> -1 <= h <= 1 && -1 <= k <= 1 && x \[Element] Reals &&
                        y \[Element] Reals && z \[Element] Reals]

(* -Log[-z + Sqrt[h^2 + k^2 - 2 h x + x^2 - 2 k y + y^2 + z^2]] - 
    Log[z + Sqrt[h^2 + k^2 - 2 h x + x^2 - 2 k y + y^2 + z^2]] + 
    Log[1 - z + Sqrt[1 + h^2 + k^2 - 2 h x + x^2 - 2 k y + y^2 - 2 z + z^2]] + 
    Log[1 + z + Sqrt[1 + h^2 + k^2 - 2 h x + x^2 - 2 k y + y^2 + 2 z + z^2]] *)

I do the second integration with the rule based integrator (Rubi) by Albert Rich (see http://www.apmaths.uwo.ca/~arich/ ), because, in contrast to Mathematica, it gives an antiderivative without discontinuities.


rint2[x_, y_, z_, h_, k_] = Int[int1, k];


Take integration values at borders to get the definite integral.


rint2def[x_, y_, z_, h_] = 
     rint2[x, y, z, h, 1] - rint2[x, y, z, h, -1] // 
      Simplify[#, Assumptions -> -1 <= h <= 1 && -1 <= k <= 1 && 
         x \[Element] Reals && y \[Element] Reals && z \[Element] Reals] &

(*   -h ArcTan[((-1 + y) (-1 + z))/((h - x) Sqrt[
2 + h^2 - 2 h x + x^2 - 2 y + y^2 - 2 z + z^2])] + 
x ArcTan[((-1 + y) (-1 + z))/((h - x) Sqrt[

2 + h^2 - 2 h x + x^2 - 2 y + y^2 - 2 z + z^2])] + 
h ArcTan[((1 + y) (-1 + z))/((h - x) Sqrt[
2 + h^2 - 2 h x + x^2 + 2 y + y^2 - 2 z + z^2])] - 
x ArcTan[((1 + y) (-1 + z))/((h - x) Sqrt[
2 + h^2 - 2 h x + x^2 + 2 y + y^2 - 2 z + z^2])] - 
h ArcTan[((1 - y) (1 + z))/((h - x) Sqrt[
2 + h^2 - 2 h x + x^2 - 2 y + y^2 + 2 z + z^2])] + 
x ArcTan[((1 - y) (1 + z))/((h - x) Sqrt[
2 + h^2 - 2 h x + x^2 - 2 y + y^2 + 2 z + z^2])] - 
h ArcTan[((1 + y) (1 + z))/((h - x) Sqrt[

2 + h^2 - 2 h x + x^2 + 2 y + y^2 + 2 z + z^2])] + 
x ArcTan[((1 + y) (1 + z))/((h - x) Sqrt[
2 + h^2 - 2 h x + x^2 + 2 y + y^2 + 2 z + z^2])] - (-1 + 
z) ArcTanh[(1 - y)/Sqrt[
2 + h^2 - 2 h x + x^2 - 2 y + y^2 - 2 z + z^2]] - 
y ArcTanh[Sqrt[2 + h^2 - 2 h x + x^2 - 2 y + y^2 - 2 z + z^2]/(
1 - z)] - 
ArcTanh[(-1 - y)/Sqrt[
2 + h^2 - 2 h x + x^2 + 2 y + y^2 - 2 z + z^2]] + 
z ArcTanh[(-1 - y)/Sqrt[

2 + h^2 - 2 h x + x^2 + 2 y + y^2 - 2 z + z^2]] + 
y ArcTanh[Sqrt[2 + h^2 - 2 h x + x^2 + 2 y + y^2 - 2 z + z^2]/(
1 - z)] + 
ArcTanh[(1 - y)/Sqrt[
2 + h^2 - 2 h x + x^2 - 2 y + y^2 + 2 z + z^2]] + 
z ArcTanh[(1 - y)/Sqrt[
2 + h^2 - 2 h x + x^2 - 2 y + y^2 + 2 z + z^2]] - 
y ArcTanh[Sqrt[2 + h^2 - 2 h x + x^2 - 2 y + y^2 + 2 z + z^2]/(
1 + z)] - 
ArcTanh[(-1 - y)/Sqrt[

2 + h^2 - 2 h x + x^2 + 2 y + y^2 + 2 z + z^2]] - 
z ArcTanh[(-1 - y)/Sqrt[
2 + h^2 - 2 h x + x^2 + 2 y + y^2 + 2 z + z^2]] + 
y ArcTanh[Sqrt[2 + h^2 - 2 h x + x^2 + 2 y + y^2 + 2 z + z^2]/(
1 + z)] - Log[-z + Sqrt[1 + h^2 - 2 h x + x^2 - 2 y + y^2 + z^2]] -
Log[-z + Sqrt[1 + h^2 - 2 h x + x^2 + 2 y + y^2 + z^2]] + 
Log[1 - z + Sqrt[2 + h^2 - 2 h x + x^2 + 2 y + y^2 - 2 z + z^2]] + 
Log[((1 - z + Sqrt[
   2 + h^2 - 2 h x + x^2 - 2 y + y^2 - 2 z + z^2]) (1 + z + Sqrt[
   2 + h^2 - 2 h x + x^2 - 2 y + y^2 + 2 z + z^2]) (1 + z + Sqrt[

   2 + h^2 - 2 h x + x^2 + 2 y + y^2 + 2 z + z^2]))/((z + Sqrt[
   1 + h^2 - 2 h x + x^2 - 2 y + y^2 + z^2]) (z + Sqrt[
   1 + h^2 - 2 h x + x^2 + 2 y + y^2 + z^2]))]   *)

The last integration has to be done numericaly.


rint3[x_, y_, z_] := NIntegrate[rint2def[x, y, z, h], {h, -1, 1}]

ContourPlot now finishes within 21 seconds.


ContourPlot[rint3[x, y, 1/2], {x, -2, 2}, {y, -2, 2}, 
             ImageSize -> 400] // Timing


enter image description here


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