Mathematica: Solving a PDE that contains Conjugate of a function or Abs of a function

Question

I'm trying numerically solve a PDE using NDSolve. I keep getting the following error:

"NDSolve::ndnum: "Encountered non-numerical value for a derivative at t == 0.."

It seems that I only get this error due to the presence of Abs[D[f[x,y,t],x]] or Conjugate[D[f[x,y,t],x]]. I created a very simple function to demonstrate this problem.

This will work:

noAbs = D[f[x, t], t] == f[x, t] f[x, t] D[f[x, t], x, x]
xrange = \[Pi]; trange = 5;
forSolve = {noAbs, f[x, 0] == Exp[I x], f[-xrange, t] == f[xrange, t]}
frul = NDSolve[forSolve, f, {x, -xrange, xrange}, {t, 0, trange}, 
   MaxStepSize -> 0.007, Method -> "MethodOfLines" ];

This won't work (note the only difference is that we don't have an Abs).

withAbs = D[f[x, t], t] == f[x, t] f[x, t] Abs[D[f[x, t], x, x]];
forSolve = {withAbs, f[x, 0] == Exp[I x], 
  f[-xrange, t] == f[xrange, t]}
frul = NDSolve[forSolve, {f}, {x, -xrange, xrange}, {t, 0, trange}, 
   MaxStepSize -> 0.007, Method -> "MethodOfLines" ];
Plot3D[Re[f[x, t]] /. frul, {x, -xrange, xrange}, {t, 0, trange}]

Right now I'm guessing Mathematica tried to take derivatives of my expressions and finds that it doesn't know how to derive Abs function. Am I right in assuming this, and how does one get around this problem?

Answer 1

With a little patience and writing everything in terms of the real and imaginary part of your function. Setting f=fRe + I fIm :

equation =  ComplexExpand[
 (Derivative[0, 1][fRe][x, t] + I Derivative[0, 1][fIm][x, t]) == 
 (fRe[x, t] + I fIm[x, t])^2  Abs[Derivative[2, 0][fRe][x, t] + I Derivative[2, 0][fIm][x, t]], 
   TargetFunctions -> {Re, Im}] ;

Initial condition :

Plot[Evaluate@solAbs[x, 0], {x, -xrange, xrange}]

Boundary condition :

Plot[Evaluate@(solAbs[-xrange, t] - solAbs[xrange, t]), {t, 0, trange}, PlotRange -> All]