[petsc-users] PetscFV and TS implicit

[Jed Brown]

Matthew Knepley <knepley at gmail.com> writes:

> I could never get the FVM stuff to make sense to me for implicit methods.
> Here is my problem understanding. If you have an FVM method, it decides
> to move "stuff" from one cell to its neighboring cells depending on the
> solution to the Riemann problem on each face, which computed the flux. This
> is
> fine unless the timestep is so big that material can flow through into the
> cells beyond the neighbor. Then I should have considered the effect of the
> Riemann problem for those interfaces. That would be in the Jacobian, but I
> don't know how to compute that Jacobian. I guess you could do everything
> matrix-free, but without a preconditioner it seems hard.

So long as we're using method of lines, the flux is just instantaneous flux, not integrated over some time step.  It has the same meaning for implicit and explicit.

An explicit method would be unstable if you took such a large time step (CFL) and an implicit method will not simultaneously be SSP and higher than first order, but it's still a consistent discretization of the problem.

It's common (done in FUN3D and others) to precondition with a first-order method, where gradient reconstruction/limiting is skipped.  That's what I'd recommend because limiting creates nasty nonlinearities and the resulting discretizations lack h-ellipticity which makes them very hard to solve.