Solving complex differential equation with ParametricNDSolveValue

Question

I am trying to solve a complex differential equation for the function $S(u,v)$ depending on the parameter $omega$. The code is:
ClearAll["Global`*"]
m = 100;
L = 2;
r[u_, v_] = 2 m (1 + ProductLog[- ((u v)/E)]);
F[u_, v_] = (32 m^3)/r[u, v]^3 Exp[-(r[u, v]/(2 m))];
Vz[u_, v_] = FullSimplify [-2 (D[F[u, v], u] D[F[u, v], v])/F[u, v] + 
    4 D[r[u, v], u, v]/r[u, v] + 2/F[u, v] D[F[u, v], u, v] + 
    2/F[u, v] D[F[u, v], u] D[r[u, v], v] + 
    2/F[u, v] D[F[u, v], v] D[r[u, v], u]];
Z[u_, v_] = Exp[-I (u + v)/2 ω] S[u, v];
sol = ParametricNDSolveValue[{D[Z[u, v], u, v] + 
     F[u, v] (L (L + 1))/r[u, v]^2 Z[u, v] + Z[u, v] Vz[u, v] == 0, 
   S[u, -1] == 1, S[1, v] == 1}, 
  S, {u, 1, 100}, {v, -100, -1}, ω]

I get the error

ParametricNDSolveValue::mconly: "For the method !("IDA"), only
machine real code is available. Unable to continue with complex values
or beyond floating-point exceptions"

So it seems that Mathematica expects real numbers, but it finds complex numbers instead. How can I solve the differential equation?

bbgodfrey · Answer

This question can be addressed by solving for Z rather than S, splitting the PDE into its real and imaginary parts, and later constructing S if desired.
solr[ω_] := NDSolveValue[{D[Z[u, v], u, v] + 
    F[u, v] (L (L + 1))/r[u, v]^2 Z[u, v] + Z[u, v] Vz[u, v] == 0, 
    Z[u, -1] == Cos[1/2 (-1 + u) ω], Z[1, v] == Cos[1/2 (1 + v) ω]}, 
    Z, {u, 1, 2}, {v, -2, -1}]
soli[ω_] := NDSolveValue[{D[Z[u, v], u, v] + 
    F[u, v] (L (L + 1))/r[u, v]^2 Z[u, v] + Z[u, v] Vz[u, v] == 0, 
    Z[u, -1] == -Sin[1/2 (-1 + u) ω], Z[1, v] == -Sin[1/2 (1 + v) ω]}, 
    Z, {u, 1, 2}, {v, -2, -1}]

zr = solr[1];
Plot3D[zr[u, v], {u, 1, 2}, {v, -2, -1}, PlotRange -> All, 
    ImageSize -> Large, AxesLabel -> {u, v, z}, LabelStyle -> {15, Black, Bold}]

zi = soli[1];
Plot3D[zi[u, v], {u, 1, 2}, {v, -2, -1}, PlotRange -> All, 
    ImageSize -> Large, AxesLabel -> {u, v, z}, LabelStyle -> {15, Black, Bold}]

Two notes.  First, the integration ranges of u and v have been greatly reduced, because the solution grows exponentially large otherwise, and Plot3D fails.  Second, using ParametricNDSolveValue instead of SetDelayed and NDSolveValue causes the kernel to crash.

Solving complex differential equation with ParametricNDSolveValue

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