[petsc-users] Local Discontinuous Galerkin with PETSc TS

Fri Mar 26 19:20:30 CDT 2021

   What is SLATE in this context?

> On Mar 23, 2021, at 2:57 PM, Matthew Knepley <knepley at gmail.com> wrote:
> 
> On Tue, Mar 23, 2021 at 11:54 AM Salazar De Troya, Miguel <salazardetro1 at llnl.gov <mailto:salazardetro1 at llnl.gov>> wrote:
> 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:
> 
> 
> I am almost certain that you do not want to do this. I am guessing the Firedrake guys will agree. Did they tell you to do this?
> If you had a large, nonlinear system for p1/p2, then a DAE would make sense. Since it is just element-wise elimination, you should
> roll it into the easy equation
> 
>   u' = H
> 
> Then you can use any integrator, as Barry says, in particular a nice symplectic integrator. My understand is that SLATE is for exactly
> this kind of thing.
> 
>   Thanks,
> 
>       Matt
>  
>                 F(U', U, t) = G(t,U)
> 
>                 p1 = f(u_x)
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>                 p2 = g(u_x)
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>                 u' - H(p1, p2) =  0
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>  
> 
> 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
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>  
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> dF/dU = [[M, 0, 0],
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>                 [0, M, 0],
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>                 [H'(p1), H'(p2), \sigma*M]]
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>  
> 
> 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.
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>  
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> Thanks
> 
> Miguel
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> From: Barry Smith <bsmith at petsc.dev <mailto:bsmith at petsc.dev>>
> Date: Monday, March 22, 2021 at 7:42 PM
> To: Matthew Knepley <knepley at gmail.com <mailto:knepley at gmail.com>>
> Cc: "Salazar De Troya, Miguel" <salazardetro1 at llnl.gov <mailto:salazardetro1 at llnl.gov>>, "Jorti, Zakariae via petsc-users" <petsc-users at mcs.anl.gov <mailto:petsc-users at mcs.anl.gov>>
> Subject: Re: [petsc-users] Local Discontinuous Galerkin with PETSc TS
> 
>  
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>  
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>    u_t  = G(u)
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>  
> 
>   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
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>  
> 
> 
> 
> 
> On Mar 22, 2021, at 7:20 PM, Matthew Knepley <knepley at gmail.com <mailto:knepley at gmail.com>> wrote:
> 
>  
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> 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)
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>                 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?
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> 
> 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.
> 
>  
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> 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.
> 
>  
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>   Thanks,
> 
>  
> 
>      Matt
> 
>  
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> 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
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> 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/ <https://urldefense.us/v3/__http:/www.cse.buffalo.edu/*knepley/__;fg!!G2kpM7uM-TzIFchu!nue-xIlrKIjtG6dGeWKiWVhSxLIOor_uLXP0UEel7pqB4YUy0y-YTHDqVX9IQCFFohVy9g$>
>  
> 
> 
> 
> -- 
> 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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