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differential equations - Improving NDSolve speed for heavily stiff problems


Having looked around the intergoogles and Mathematica.SE, I thought I'd pose a question with a minimum working example.



Here is the situation I am trying to improve:



  1. I am solving a 4th order non linear PDE with NDSolve.

  2. It is stiff and I use a stiff solver such as BDF or LSODA.

  3. On occassion, I have no choice but to increase the MaxStepFraction to uncomfortable levels.

  4. As a result, the code runs longer than usual (made worse by the fact that it is a stiff equation to begin with)


Is there any way I could improve NDSolve performance/speed?


Here is my minimum example:




$HistoryLength = 0;
Needs["VectorAnalysis`"]
Needs["DifferentialEquations`InterpolatingFunctionAnatomy`"];
Clear[Eq0, EvapThickFilm, h, Bo, \[Epsilon], K1, \[Delta], Bi, m, r]
Eq0[h_, {Bo_, \[Epsilon]_, K1_, \[Delta]_, Bi_, m_, r_}] := \!\(
\*SubscriptBox[\(\[PartialD]\), \(t\)]h\) + 
    Div[-h^3 Bo Grad[h] + 
      h^3 Grad[Laplacian[h]] + (\[Delta] h^3)/(Bi h + K1)^3 Grad[h] + 
      m (h/(K1 + Bi h))^2 Grad[h]] + \[Epsilon]/(
    Bi h + K1) + (r) D[D[(h^2/(K1 + Bi h)), x] h^3, x] == 0;

SetCoordinates[Cartesian[x, y, z]];
EvapThickFilm[Bo_, \[Epsilon]_, K1_, \[Delta]_, Bi_, m_, r_] := 
  Eq0[h[x, y, t], {Bo, \[Epsilon], K1, \[Delta], Bi, m, r}];
TraditionalForm[EvapThickFilm[Bo, \[Epsilon], K1, \[Delta], Bi, m, r]];
L = 2*92.389;

TMax = 3100*100;
Off[NDSolve::mxsst];
Clear[Kvar];
Kvar[t_] :=  Piecewise[{{1, t <= 1}, {2, t > 1}}]

(*Ktemp = Array[0.001+0.001#^2&,13]*)
hSol = h /. NDSolve[{
      (*Bo,\[Epsilon],K1,\[Delta],Bi,m,r*)

      EvapThickFilm[0.003, 0, 1, 0, 1, 0.025, 0],
      h[0, y, t] == h[L, y, t],
      h[x, 0, t] == h[x, L, t],
      (*h[x,y,0] == 1.1+Cos[x] Sin[2y] *)

      h[x, y, 0] == 

       1 + (-0.25 Cos[2 \[Pi] x/L] - 0.25 Sin[2 \[Pi] x/L]) Cos[
          2 \[Pi] y/L]
      },
     h,
     {x, 0, L},
     {y, 0, L},
     {t, 0, TMax},
     Method -> {"BDF", "MaxDifferenceOrder" -> 1},
     MaxStepFraction -> 1/50
     ][[1]] // AbsoluteTiming


A BDF limited to Order 1 needs about 41 seconds to solve the equation until ****failure**** while the LSODA allowed up to order 12 does a fantastic job of cutting it down to 18 seconds.



Now when I increase the MaxStepFraction, I obviously increase the grid density. I am currently running a case that has several thousand grid points and has taken 24+ HOURS, yes hours and hasn't given me a solution yet. This was expected as I have run cases that took about 3-4 hours before with a coarser grid and do hog the ram (they take up about ~3-4GBs mostly because I am exporting data as .MAT files)


What suggestions could be provided to improve the speed for such a stiff equation?


I have also tried stopping tests and it doesn't quite help all the time as I'd rather mathematica quit my program naturally as a result of overbearing stiffness than artificially through a stopping test. (The former has physical significance)


Yes, this question bears resemblance to this but I don't think its the same.


I have given Parallelize a thought but it doesn't work on NDSolve. Any options that I have either on the Mathematica front, computing front, or saving the interpolation function data?



Edit:



Using the LaunchKernel[n] option just before the NDSolve cell didn't do much. My AbsoluteTiming barely even changed.


CloseKernels[];
LaunchKernels[3];
L = 2*92.389; TMax = 3100*100;
.........
......

Edit 2:


By using Table and launching up to 6 kernels, these are the results that I got:




{{1,{19.454883,InterpolatingFunction[{{0.,184.778},{0.,184.778},{0.,282761.}},<>]}}, {2,{19.162008,InterpolatingFunction[{{0.,184.778},{0.,184.778},{0.,282761.}},<>]}}, {3,{18.919101,InterpolatingFunction[{{0.,184.778},{0.,184.778},{0.,282761.}},<>]}}, {4,{20.166785,InterpolatingFunction[{{0.,184.778},{0.,184.778},{0.,282761.}},<>]}}, {5,{20.265163,InterpolatingFunction[{{0.,184.778},{0.,184.778},{0.,282761.}},<>]}}, {6,{20.556365,InterpolatingFunction[{{0.,184.778},{0.,184.778},{0.,282761.}},<>]}}}



So with more kernels, the performance actually degraded....? Wha...?



Answer



Yes, it is stiff -- but the main issue that I see is that the solution goes wild near the TMax that you specify. That's because you need a super-fine spatial grid to accurately represent what happens when the higher order terms finally manifest themselves. It's going to take a lot of time and a lot of memory (MinPoints option), and there's no way around it.


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