Merge Jacobian coloring into PETSc solver

[bendudson]

Takes the coloring code from the snes/beuler solver, and merges it into the petsc solver. This uses the PETSc TS interface, enabling use of many time-integration algorithms, including sundials/cvode.

PETSc TS manual: https://petsc.org/release/manual/ts/

Use with e.g. solver:type=petsc solver:ts_type=beuler

Other ts_type choices include bdf, pseudo (https://petsc.org/release/manualpages/TS/TSPSEUDO/)
See list of methods here: https://petsc.org/release/overview/integrator_table/#integrator-table

Other solver settings include:

• adapt_type, that sets the adaptive timestep method https://petsc.org/release/manualpages/TS/TSAdaptType/
• snes_type, the nonlinear solver method
• ksp_type, the linear solver method
• pc_type, the preconditioner method

use e.g. petsc:ts_monitor, petsc:snes_monitor to print diagnostics; petsc:help to print the available PETSc options.

The combination of bdf method and STRUMPACK direct solver seems to be quite effective:

[solver]
type = petsc
ts_type = bdf
ksp_type = preonly
pc_type = lu

[petsc]
pc_factor_mat_solver_type = strumpack

That solves in 2 1/2 minutes a 2D transport problem that took CVODE (without preconditioning) 2 1/2 hours.

If PETSc is compiled with sundials (configure flag --download-sundials) then the Jacobian coloring and PETSc preconditioners can be used with CVODE (2.5.0):

[solver]
type = petsc
ts_type = sundials   # Use the CVODE solver through PETSc
pc_type = lu             # Jacobian coloring, then form preconditioner using LU
start_timestep = 0.01 # Otherwise fails on first step

[petsc]
pc_factor_mat_solver_type = strumpack

This solved the same 2D transport problem in 1 1/2 minutes.

There are many (many!) settings to try and tune.