[petsc-users] Local Discontinuous Galerkin with PETSc TS

[Salazar De Troya, Miguel]

The calculation of p1 and p2 are done by solving an element-wise local problem using u^n. I guess I could embed this calculation inside of the calculation for G = H(p1, p2). However, I am hoping to be able to solve the problem using firedrake-ts so the formulation is all clearly in one place and in variational form. Reading the manual, Section 2.5.2 DAE formulations, the Hessenberg Index-1 DAE case seems to be what I need, although it is not clear to me how one can achieve this with an IMEX scheme. If I have:

                F(U', U, t) = G(t,U)
                p1 = f(u_x)
                p2 = g(u_x)
                u' - H(p1, p2) =  0

where U = (p1, p2, u), F(U’, U, t) = [p1, p2, u’ - H(p1, p2)],] and G(t, U) = [f(u_x), g(u_x), 0], is there a solver strategy that will solve for p1 and p2 first and then use that to solve the last equation? The jacobian for F in this formulation would be

dF/dU = [[M, 0, 0],
                [0, M, 0],
                [H'(p1), H'(p2), \sigma*M]]

where M is a mass matrix, H'(p1) is the jacobian of H(p1, p2) w.r.t. p1 and H'(p2), the jacobian of H(p1, p2) w.r.t. p2. H'(p1) and H'(p2) are unnecessary for the solver strategy I want to implement.

Thanks
Miguel



From: Barry Smith <bsmith at petsc.dev>
Date: Monday, March 22, 2021 at 7:42 PM
To: Matthew Knepley <knepley at gmail.com>
Cc: "Salazar De Troya, Miguel" <salazardetro1 at llnl.gov>, "Jorti, Zakariae via petsc-users" <petsc-users at mcs.anl.gov>
Subject: Re: [petsc-users] Local Discontinuous Galerkin with PETSc TS


   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<mailto: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://urldefense.us/v3/__https:/www.sciencedirect.com/science/article/pii/S0021999110005255__;!!G2kpM7uM-TzIFchu!nue-xIlrKIjtG6dGeWKiWVhSxLIOor_uLXP0UEel7pqB4YUy0y-YTHDqVX9IQCHtstz33g$>. 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


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