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 <[email protected]> wrote: > > On Mon, Mar 22, 2021 at 7:53 PM Salazar De Troya, Miguel via petsc-users > <[email protected] <mailto:[email protected]>> 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/>
