[petsc-users] Local Discontinuous Galerkin with PETSc TS

[Barry Smith]

   u_t  = G(u)

  I don't see why you won't just compute any needed u_x from the given u and then you can use any explicit or implicit TS solver trivially. For implicit methods it can automatically compute the Jacobian of G for you or you can provide it directly. Explicit methods will just use the "old" u while implicit methods will use the new.

  Barry


> On Mar 22, 2021, at 7:20 PM, Matthew Knepley <knepley at gmail.com> wrote:
> 
> On Mon, Mar 22, 2021 at 7:53 PM Salazar De Troya, Miguel via petsc-users <petsc-users at mcs.anl.gov <mailto: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 <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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