[petsc-users] Solving advection equations implicitly

[Jed Brown]

There are not unconditionally strongly stable integrators of order
higher than 1, unless you look into downwind methods which present more
grave solver challenges.  You can use implicit Euler, but it will be
very diffusive.  You can test this out using a direct solve, but I think
you'll be disappointed in the results of this discretization, in which
case fast preconditioning is irrelevant.

Nishant Nangia <nishantnangia329 at gmail.com> writes:

> We basically want something that is stable and non-oscillatory for any
> given time step size.
>
> A little more context: Q is a fluid density variable, which we are
> currently updating using an explicit forward Euler step. This updated
> quantity is later used in a conservative discretization of the
> Navier-Stokes (NS) equations on a staggered mesh. We are using finite
> volume/finite differences.
>
> We observe oscillations in Q after a few time steps, which causes the
> overall mass of the domain to change and breaks the linear solvers for NS.
> We are experimenting with some strong stability preserving, TVD schemes to
> update this density, but were thinking of trying an implicit update instead.
>
>
> *Nishant Nangia*
> Northwestern University
> Ph.D. Candidate | Engineering Sciences and Applied Mathematics
> Tech L386
>
> On Mon, May 7, 2018 at 4:56 PM, Jed Brown <jed at jedbrown.org> wrote:
>
>> Do you want it to be time accurate (implies CFL number is modest) or do
>> you want very large time steps?  If very large time steps, why not
>> steady state?
>>
>> Nishant Nangia <nishantnangia329 at gmail.com> writes:
>>
>> > Hi all,
>> >
>> > I want to implicitly solve a linear advection equation of the form:
>> > dQ/dt + div(u*Q) = 0
>> >
>> > for a scalar quantity Q, with some known velocity field u. Note that it
>> is
>> > purely advection with no diffusion term.
>> >
>> > Is there a recommended solver/preconditioner combination to solve
>> something
>> > like this?
>> >
>> > *Nishant Nangia*
>> > Northwestern University
>> > Ph.D. Candidate | Engineering Sciences and Applied Mathematics
>> > Tech L386
>>