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differential equations - Need help in plotting a function with two variables in ODE


The solution of this ODE was given in this link


Here I am asking, if L is a function of tot, g, Z and tot depends on Z and g, which means I need to solve the ODE for each value of Z and g to get tot and then pass it to L function, after that Plot L.

I am not sure should I use ContourPlot or somethings els. Help please.


p[Z0_, g0_, k0_, R0_] := 
  Block[{Z = Z0, 
         g = Rationalize[g0, 0],
         k2 = Rationalize[k0, 0],
         ϵ = 10^-4, 
         R = Rationalize[R0, 0]},
    ps = 
      ParametricNDSolveValue[
        {y''[r] + 2 y'[r]/r == k2 Sinh[y[r]], 

         y[ϵ] == y0, y'[ϵ] == 0, 
         WhenEvent[r == 1, y'[r] -> y'[r] + Z g]}, 
        {y, y'}, {r, ϵ, R}, {y0}, 
        Method -> "StiffnessSwitching", 
        WorkingPrecision -> 20];

sol = FindRoot[Last[ps[y0]][R], {y0, -1}, Evaluated -> False][[1, 2]];       
tot = 4 π sol NIntegrate[r^2  Exp[-First[ps[sol]][r]], {r, 0, R}];
L= Z/g*tot;


ContourPlot[L, {g, 0.02, 0.06}, {Z, 500, 800}]]

Quiet[
  Table[p[Z, g, 0.0002, 1.5], {g, 0.02, 0.06, .02},{Z, 500, 800, 200}]]

Answer



The desired contour plot can be obtained as follows. Begin with a slight modification to the code in the question.


p[Z0_, g0_, k0_, R0_] := Block[{Z = Z0, 
  g = Rationalize[g0, 0], k2 = Rationalize[k0, 0], ϵ = 10^-4, R = Rationalize[R0, 0]}, 
  ps = ParametricNDSolveValue[{y''[r] + 2 y'[r]/r == k2 Sinh[y[r]], y[ϵ] == y0, y'[ϵ] == 0, 
    WhenEvent[r == 1, y'[r] -> y'[r] + Z g]}, {y, y'[R]}, {r, ϵ], R}, {y0}, 

    Method -> "StiffnessSwitching", WorkingPrecision -> 20];
  sol = FindRoot[Last[ps[y0]], {y0, -1}, Evaluated -> False][[1, 2]];
  tot = 4 π sol NIntegrate[r^2 Exp[-First[ps[sol]][r]], {r, ϵ, R}];
  L = Z/g*tot]

Specifically, {y, y'} is replaced by {y, y'[R]} to save a bit of time, because the value of y' is needed only at r = R; a corresponding change is made in the first argument of FindRoot; the lower bound of NIntegrate is set to ϵ, because ps is undefined for smaller values of r; and ContourPlot is removed from within the definition of p.


The desired contour plot then is


ContourPlot[Quiet[p[Z, g, 0.0002, 1.5]], {g, 0.02, 0.06}, {Z, 500, 800}, 
    PlotPoints -> {5, 4}, MaxRecursion -> 1, PlotLegends -> Automatic, 
    FrameLabel -> {g, Z}, ImageSize -> Large, LabelStyle -> {Bold, Black, 15}]


enter image description here


which takes about two minutes on my computer. The plotting can, however, be parallelized with equal resolution using


DistributeDefinitions[p];
tab = ParallelTable[Quiet[p[Z, g, 0.0002, 1.5]], {Z, 500, 800, 50}, {g, 0.02, 0.06, .005}];
ListContourPlot[tab, DataRange -> {{.02, .06}, {500, 800}}, PlotLegends -> Automatic, 
    FrameLabel -> {g, Z}, ImageSize -> Large, LabelStyle -> {Bold, Black, 15}]

which produces the same plot in 37 seconds. (My computer has six processors.) Finally, the table requested in the question is


Table[Quiet[p[Z, g, 0.0002, 1.5]], {g, 0.02, 0.06, .02}, {Z, 500, 800, 200}]

(* {{-3.41941*10^11, -6.80714*10^11}, {-3.51627*10^11, -6.9574*10^11}, 
    {-3.55499*10^11, -7.0144*10^11}} *)

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