Follow up: how to plot a function under NDSolve domain

Question

Here is the details of the code to plot.

Clear[x, y, ψ1, ψ2, ψ3, ψ4, eqn, eqnWithInitial,v, j];
eqn = Thread[
       I D[{ψ1[x, y, t], ψ2[x, y, t], ψ3[x, y, t], ψ4[
            x, y, t]}, 
          t] == {v (-I D[ψ3[x, y, t], x] - D[ψ3[x, y, t], y]) + 
          2 Δ ψ4[x, y, t], 
         v (-I D[ψ4[x, y, t], x] - D[ψ4[x, y, t], y]), 
         v (-I D[ψ1[x, y, t], x] + D[ψ1[x, y, t], y]), 
         v (-I D[ψ2[x, y, t], x] + D[ψ2[x, y, t], y]) + 
          2 Δ ψ1[x, y, t]}];
eqnWithInitial = 
      Join[eqn, 
       Thread[{ψ1[x, y, 0], ψ2[x, y, 0], ψ3[x, y, 
           0], ψ4[x, y, 0]} == {1, 1, 1, 
           1} (x + I*y) Exp[-(x^2 + y^2)]], 
       Thread[{ψ1[-5, y, t], ψ2[-5, y, t], ψ3[-5, y, 
           t], ψ4[-5, y, t]} == {ψ1[5, y, t], ψ2[5, y, 
           t], ψ3[5, y, t], ψ4[5, y, t]}], 
       Thread[{ψ1[x, -5, t], ψ2[x, -5, t], ψ3[x, -5, 
           t], ψ4[x, -5, t]} == {ψ1[x, 5, t], ψ2[x, 5, 
           t], ψ3[x, 5, t], ψ4[x, 5, t]}]];

v = 1;
Δ = 1;
tMax = 8;

solution = 
      First @ NDSolve[
        eqnWithInitial, {ψ1[x, y, t], ψ2[x, y, t], ψ3[x, y,
           t], ψ4[x, y, t]}, {x, -5, 5}, {y, -5, 5}, {t, 0, tMax}, 
        Method -> {"MethodOfLines", 
          "SpatialDiscretization" -> {"TensorProductGrid", 
            "DifferenceOrder" -> "Pseudospectral"}}];

Ψ1[x_, y_, t_] = ψ1[x, y, t] /. solution;
Ψ2[x_, y_, t_] = ψ2[x, y, t] /. solution;
Ψ3[x_, y_, t_] = ψ3[x, y, t] /. solution;
Ψ4[x_, y_, t_] = ψ4[x, y, t] /. solution; 

myrotorz1[x_?NumericQ, y_?NumericQ, t_?NumericQ] = 
  I/2 * (-Conjugate[D[Ψ1[x, y, t], y]] * D[Ψ1[x, y, t], x] + 
    D[Ψ1[x, y, t],y] * Conjugate[D[Ψ1[x, y, t], x]] + 
    Conjugate[D[Ψ1[x, y, t], x]] * D[Ψ1[x, y, t], y] - 
    D[Ψ1[x, y, t], x] * Conjugate[D[Ψ1[x, y, t], y]]);

Here is something that may be reasonable as R. M. suggested

NIntegrate[myrotorz1[x, y, 0], {x, -5, 5}, {y, -5, 5},Method -> "Trapezoidal"]

(Debug) During evaluation of In[18]:= NIntegrate::ncvi: NIntegrate failed to converge to prescribed accuracy after 9 iterated refinements in y in the region {{-5.,5.},{-5.,5.}}. NIntegrate obtained 0.012104690946256463` and 0.8391566465710514` for the integral and error estimates. >>

(Debug) Out[18]= 0.0121047

My point is to plot for myrotorz1[x, y, t] as a function of t but takes too much time. Any tips would be greatly appreciated.

Plot[NIntegrate[myrotorz1[x, y, t], {x, -5, 5}, {y, -5, 5},Method -> "Trapezoidal"], {t, 0, 4}]

Answer 1

Nothing much can be said without knowing the form of myrotorz1[x, y, t] I would first include NumericQ to the function. But for a simple function you can do it easily. Considering a simple function, I will do the following.

f[x_, y_, t_] := x^2 + y^2 + t^2
data[t_] := NIntegrate[f[x, y, t], {x, 0, 1}, {y, 0, 1}]
toplot = ParallelTable[data[t], {t, 0, 10}] // ListPlot