How correctly use WhenEvent to detect when gradient is over a certain value?

Question

So I wish to stop integrating my PDE when the spatial gradient is larger than some value, let's say 10. That is to say, my WhenEvent condition wants to be stop when the the maximum gradient over all x at a given t exceeds a threshold.

However, when I try implement the code I get "The function value is not True or False errors".

Here's my attempt:

NDSolve[{Derivative[0, 1][h][x, t] + 2000 h[x, t]*Derivative[1, 0][h][x, t] == 
  1/4 D[(1 - 4 h[x, t]^2)^3 Derivative[1, 0][h][x, t], x], 
  h[-1/2, t] == 0, h[1/2, t] == 0, h[x, 0] == 0.1 Sin[8 π (x - 1/2)], 
  WhenEvent[NMaximize[Derivative[1, 0][h][x, t]] > 10, tEnd = t; 
  "StopIntegration"]}, h[x, t], {x, -1/2, 1/2}, {t, 0, 10}, 
  Method -> {"MethodOfLines", "DiscretizedMonitorVariables" -> True, 
  "SpatialDiscretization" -> {"TensorProductGrid", "MinPoints" -> 100000}}];

Any ideas what I'm doing wrong?

Answer 1

After playing with this a bit: it seems WhenEvent[ h[x,t] ] gets passed the list of discrete x-grid values at each time step. So we can numerically compute the gradient:

rr = Reap[
   sol = NDSolve[{...,
      WhenEvent[
       Sow[Max[Abs[Subtract @@@ Partition[ h[x, t] , 2, 1]]] 100000] > 10, 
       "StopIntegration"]},
     h[x, t], {x, -1/2, 1/2}, {t, 0, 10},
     Method -> {"MethodOfLines", 
       "DiscretizedMonitorVariables" -> True, 
       "SpatialDiscretization" -> {"TensorProductGrid", 
         "MinPoints" -> 100000}}]];
 f = h[x, t] /. First@sol;

note I assume uniform grid spacing which in this case is set by your specified minpoints (But that's not guaranteed). You can verify that by looking at the interpolation function grid.

The critical time:

tmax = f[[0]]["Domain"][[2, 2]]

0.000172062

Plot3D[f, {x, -1/2, 1/2}, {t, 0, tmax}]

here is the derivative and you can see it becoming unstable..

g = D[f, x];
Plot3D[g, {x, -1/2, 1/2}, {t, 0, tmax}, PlotRange -> {-10, 10}]

here is the max gradient value captured by Reap/Sow

ListPlot[rr[[2, 1]]]