[petsc-users] Local Discontinuous Galerkin with PETSc TS

[Matthew Knepley]

On Mon, Mar 22, 2021 at 7:53 PM Salazar De Troya, Miguel via petsc-users <
petsc-users at mcs.anl.gov> wrote:

> Hello
>
>
>
> I am interested in implementing the LDG method in “A local discontinuous
> Galerkin method for directly solving Hamilton–Jacobi equations”
> https://www.sciencedirect.com/science/article/pii/S0021999110005255. The
> equation is more or less of the form (for 1D case):
>
>                 p1 = f(u_x)
>
>                 p2 = g(u_x)
>
>                 u_t  = H(p1, p2)
>
>
>
> where typically one solves for p1 and p2 using the previous time step
> solution “u” and then plugs them into the third equation to obtain the next
> step solution. I am wondering if the TS infrastructure could be used to
> implement this solution scheme. Looking at the manual, I think one could
> set G(t, U) to the right-hand side in the above equations and F(t, u, u’) =
> 0 to the left-hand side, although the first two equations would not have
> time derivative. In that case, how could one take advantage of the operator
> split scheme I mentioned? Maybe using some block preconditioners?
>

Hi Miguel,

I have a simple-minded way of understanding these TS things. My heuristic
is that you put things in F that you expect to want
at u^{n+1}, and things in G that you expect to want at u^n. It is not that
simple, since you could for instance move F and G
to the LHS and have Backward Euler, but it is my rule of thumb.

So, were you looking for an IMEX scheme? If so, which terms should be
lagged? Also, from the equations above, it is hard to
see why you need a solve to calculate p1/p2. It looks like just a forward
application of an operator.

  Thanks,

     Matt


> I am trying to solve the Hamilton-Jacobi equation u_t – H(u_x) = 0. I
> welcome any suggestion for better methods.
>
>
>
> Thanks
>
> Miguel
>
>
>
> Miguel A. Salazar de Troya
>
> Postdoctoral Researcher, Lawrence Livermore National Laboratory
>
> B141
>
> Rm: 1085-5
>
> Ph: 1(925) 422-6411
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener

https://www.cse.buffalo.edu/~knepley/ <http://www.cse.buffalo.edu/~knepley/>
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