[petsc-users] TS and snes_grid_sequence

[Hong Zhang]

There is one particular integration method that may make the life easier — backward Euler. It has only one stage at each time step.

To Francesco: which TS method are you using? Is it backward Euler or the default theta method? The default one is not stiffly accurate, thus not good for stiff problems.

Hong 
 
> On Feb 10, 2016, at 2:06 PM, Barry Smith <bsmith at mcs.anl.gov> wrote:
> 
> 
>   Francesco,
> 
>     Unfortunately the -snes_grid_sequence option only works with a single SNES nonlinear solve and not directly within an outer time-integration scheme.
> 
>     We do not currently have library code to do grid sequence within a time-step. I believe that providing this requires a great deal of "replumbing" of the TS solvers to make them "grid sequence aware". This is true I believe for both traditional grid sequencing and FAS (I would be happy if Emil or Jed could prove me wrong).  There are seemingly multiple ways to do the grid sequencing: 
> 
> 1) Restrict the current solution to a coarse mesh and do a single stage of the ODE integrator (which means solve a single nonlinear solve) on that coarse mesh, interpolate the solution to the next mesh and use it as an initial guess for the single stage on that mesh and then continue to up the finest mesh. Note that since the specific ODE integrator determines the "recipe" for the SNES function evaluation this is not something that can be done in pure user code, it must use the "plumbing" of the specific ODE integrator.
> 
> an alternative is
> 2) Restrict the current solution to a coarse mesh and do the entire timestep of the ODE integrator (which means multiple stage and multiple nonlinear solvers) saving all the intermediate solutions (for each stage). Interpolate all the intermediate solutions (as well as the final solution) to the finer mesh and repeat the entire timestep computation but using the interpolated intermediate solutions as the initial guesses for the nonlinear solves). Continue to the finest mesh. Again this cannot be done in user code and requires replumbing the ODE integrator.
> 
>   I think 2 will produce different "results" from 1 since for 1 the coarse grid initial values on (for example) the second stage will be obtained from restricting the finest grid solution from the first stage while with 2 the initial values on the second stage will be directly the coarse solution from the first stage.
> 
> To do the FAS is even more involved, I can only conceive of doing it for a single stage. Again the FAS has to be plumbed directly into the specific ODE solver to define the correct functions for the various levels.
> 
> With the above approaches the coarser nonlinear problems should be "easier" than the finest because the time-step remains small but the space discretization is larger. But the only way the approaches can work is if the interpolated coarse grid solution used on the finer mesh is within the basin of attraction on that mesh, otherwise you will still have the same problem you have now of lack of convergence.
> 
> I don't know if anyone actually does this and how much it helps or could help. I welcome input from others.
> 
> Based on your convergence experience it sounds like your solution is changing rather rapidly in time? Is the solution somewhat smooth in space? If not then I don't think grid sequencing will help much. 
> 
> 
>  Barry
> 
> 
> 
> 
> 
>> On Feb 10, 2016, at 9:38 AM, Francesco Magaletti <francesco.magaletti at uniroma1.it> wrote:
>> 
>> Hi everyone,
>> 
>> I’m trying to solve a system of time dependent PDE’s, strongly non linear and quite stiff. The system is 1D, but I need a large number of grid points (1-10 million points). I’m facing some convergence problem of the Newton solver: if I use a small timestep (10^-5) the SNES converges correctly, but if I increase it, the Newton method diverges. Moreover, when I increase the number of grid points, the timestep for Newton convergence decreases. Such a restriction on the timestep is a big problem for my simulations.
>> 
>> I thought that the option -snes_grid_sequence could help the SNES to converge, but when I run the code the first grid converges and then the simulations abort with errors. Here is an example with 60000grid points:
>> 
>> mpiexec -n 4 ./code_petsc -ts_monitor -snes_monitor -snes_converged_reason -ksp_converged_reason -ftime 8.0 -ksp_rtol 1.0e-8 -snes_atol 1.0e-12 -snes_rtol 1.0e-14 -snes_stol 1.0e-12 -snes_max_it 2000 -dt 2.0e-4 -nx 60000 -snes_grid_sequence 1
>> 
>> 0 TS dt 0.0002 time 0
>>     0 SNES Function norm 4.597340391780e+04 
>>     Linear solve converged due to CONVERGED_RTOL iterations 9
>>     1 SNES Function norm 3.986169680241e+03 
>>     Linear solve converged due to CONVERGED_RTOL iterations 14
>>     2 SNES Function norm 4.292840674611e+01 
>>     Linear solve converged due to CONVERGED_RTOL iterations 15
>>     3 SNES Function norm 1.044935520416e-02 
>>     Linear solve converged due to CONVERGED_RTOL iterations 14
>>     4 SNES Function norm 2.936744161136e-06 
>>     Linear solve converged due to CONVERGED_RTOL iterations 10
>>     5 SNES Function norm 4.471992600451e-08 
>>   Nonlinear solve converged due to CONVERGED_SNORM_RELATIVE iterations 5
>> [0]PETSC ERROR: --------------------- Error Message --------------------------------------------------------------
>> [0]PETSC ERROR:   
>> [0]PETSC ERROR: Must call TSSetRHSFunction() and / or TSSetIFunction()
>> [0]PETSC ERROR: See http://www.mcs.anl.gov/petsc/documentation/faq.html for trouble shooting.
>> [0]PETSC ERROR: Petsc Release Version 3.6.3, Dec, 03, 2015 
>> [0]PETSC ERROR: Configure options --with-cc=gcc --with-cxx=g++ --with-fc=gfortran --download-fblaslapack --download-mpich --download-mpich-device=ch3:nemesis
>> [0]PETSC ERROR: #1 TSComputeIFunction() line 723 in /home/magaletto/Scaricati/petsc-3.6.3/src/ts/interface/ts.c
>> [0]PETSC ERROR: #2 SNESTSFormFunction_Theta() line 483 in /home/magaletto/Scaricati/petsc-3.6.3/src/ts/impls/implicit/theta/theta.c
>> [0]PETSC ERROR: #3 SNESTSFormFunction() line 4138 in /home/magaletto/Scaricati/petsc-3.6.3/src/ts/interface/ts.c
>> [0]PETSC ERROR: #4 SNESComputeFunction() line 2067 in /home/magaletto/Scaricati/petsc-3.6.3/src/snes/interface/snes.c
>> [0]PETSC ERROR: #5 SNESSolve_NEWTONLS() line 184 in /home/magaletto/Scaricati/petsc-3.6.3/src/snes/impls/ls/ls.c
>> [0]PETSC ERROR: #6 SNESSolve() line 3906 in /home/magaletto/Scaricati/petsc-3.6.3/src/snes/interface/snes.c
>> [0]PETSC ERROR: #7 TSStep_Theta() line 198 in /home/magaletto/Scaricati/petsc-3.6.3/src/ts/impls/implicit/theta/theta.c
>> [0]PETSC ERROR: #8 TSStep() line 3101 in /home/magaletto/Scaricati/petsc-3.6.3/src/ts/interface/ts.c
>> [0]PETSC ERROR: #9 TSSolve() line 3285 in /home/magaletto/Scaricati/petsc-3.6.3/src/ts/interface/ts.c
>> 
>> ===================================================================================
>> =   BAD TERMINATION OF ONE OF YOUR APPLICATION PROCESSES
>> =   EXIT CODE: 6
>> 
>> 
>> Where is the mistake? 
>> Could you suggest a "more convergent” SNES? Maybe a FAS as a nonlinear preconditioner? 
>> 
>> Hope someone can help here.
>> best regards
>> 
>> Francesco
>