Plotting with NDSolve`StateData`

Question

This question is a continuation of this.

As answered by J.M., I am able to iterate a solution. However, I am unable to plot it. I am sure that this has to do with my inability to understand how the plot routine works and I can't seem to figure it out:

state = First[
  NDSolve`ProcessEquations[{D[u[t, x], t] == D[u[t, x], x, x], 
    u[0, x] == 0, u[t, 0] == Sin[t], u[t, 5] == 0}, u, t, {x, 0, 5}]]

NDSolve`Iterate[state, 5]
NDSolve`ProcessSolutions[state, "Forward"]

Plot3D[
 Evaluate[u[t, x] /. NDSolve`ProcessSolutions[state, "Forward"]],
 {t,0,2},
 {x, 0, 5}
 ]

gives me a blank plot with no curves. What am I doing wrong? I can't omit {t, 0, 2} as that would give me an error.

Answer 1

If you look at the interpolation object that's returned, it's not a function u(t,x) only u(x) with fixed t. In fact, it returns both u[x] and u'[x] for fixed t. So try

state = First[
NDSolve`ProcessEquations[{D[u[t, x], t] == D[u[t, x], x, x], 
u[0, x] == 0, u[t, 0] == Sin[t], u[t, 5] == 0}, u, t, {x, 0, 5}]]

Table[NDSolve`Iterate[state, t];
Plot[NDSolve`ProcessSolutions[state, "Forward"][[All, 2]], {x, 0, 5},
PlotRange -> 1], {t, 0, 5}]

Answer 2

Something like this?

L = -10;
state = First[
   NDSolve`ProcessEquations[{D[u[t, x], 
       t] == + D[u[t, x], x, x] - Sin[u[t, x]],
     u[0, x] == Exp[-(x^2)],
     u[t, -L] == u[t, L]}, u, t, {x, -L, L}, 
    Method -> {"MethodOfLines", 
      "SpatialDiscretization" -> {"TensorProductGrid", 
        "DifferenceOrder" -> "Pseudospectral"}}]];
GraphicsGrid[Partition[Table[
   NDSolve`Iterate[state, tau ];

   Plot[Evaluate[
     u[tau, x] /. 
      NDSolve`ProcessSolutions[state, "Forward"]], {x, -L, L}, 
    PlotRange -> {-1/4, 1/4}],
   {tau, 0., 3., .5}], 2]]