What is SLATE in this context?
> On Mar 23, 2021, at 2:57 PM, Matthew Knepley <knep...@gmail.com> wrote:
>
> On Tue, Mar 23, 2021 at 11:54 AM Salazar De Troya, Miguel
> <salazardet...@llnl.gov <mailto:salazardet...@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)
>
> 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 <bsm...@petsc.dev <mailto:bsm...@petsc.dev>>
> Date: Monday, March 22, 2021 at 7:42 PM
> To: Matthew Knepley <knep...@gmail.com <mailto:knep...@gmail.com>>
> Cc: "Salazar De Troya, Miguel" <salazardet...@llnl.gov
> <mailto:salazardet...@llnl.gov>>, "Jorti, Zakariae via petsc-users"
> <petsc-users@mcs.anl.gov <mailto:petsc-users@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 <knep...@gmail.com
> <mailto:knep...@gmail.com>> wrote:
>
>
>
> On Mon, Mar 22, 2021 at 7:53 PM Salazar De Troya, Miguel via petsc-users
> <petsc-users@mcs.anl.gov <mailto:petsc-users@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
>
>
>
>
>
> --
>
> 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/>
