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 <[email protected]>
Date: Monday, March 22, 2021 at 7:42 PM
To: Matthew Knepley <[email protected]>
Cc: "Salazar De Troya, Miguel" <[email protected]>, "Jorti, Zakariae via
petsc-users" <[email protected]>
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
<[email protected]<mailto:[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://urldefense.us/v3/__https:/www.sciencedirect.com/science/article/pii/S0021999110005255__;!!G2kpM7uM-TzIFchu!nue-xIlrKIjtG6dGeWKiWVhSxLIOor_uLXP0UEel7pqB4YUy0y-YTHDqVX9IQCHtstz33g$>.
The equation is more or less of the form (for 1D case):
p1 = f(u_x)
p2 = g(u_x)
u_t = H(p1, p2)
where typically one solves for p1 and p2 using the previous time step solution
“u” and then plugs them into the third equation to obtain the next step
solution. I am wondering if the TS infrastructure could be used to implement
this solution scheme. Looking at the manual, I think one could set G(t, U) to
the right-hand side in the above equations and F(t, u, u’) = 0 to the left-hand
side, although the first two equations would not have time derivative. In that
case, how could one take advantage of the operator split scheme I mentioned?
Maybe using some block preconditioners?
Hi Miguel,
I have a simple-minded way of understanding these TS things. My heuristic is
that you put things in F that you expect to want
at u^{n+1}, and things in G that you expect to want at u^n. It is not that
simple, since you could for instance move F and G
to the LHS and have Backward Euler, but it is my rule of thumb.
So, were you looking for an IMEX scheme? If so, which terms should be lagged?
Also, from the equations above, it is hard to
see why you need a solve to calculate p1/p2. It looks like just a forward
application of an operator.
Thanks,
Matt
I am trying to solve the Hamilton-Jacobi equation u_t – H(u_x) = 0. I welcome
any suggestion for better methods.
Thanks
Miguel
Miguel A. Salazar de Troya
Postdoctoral Researcher, Lawrence Livermore National Laboratory
B141
Rm: 1085-5
Ph: 1(925) 422-6411
--
What most experimenters take for granted before they begin their experiments is
infinitely more interesting than any results to which their experiments lead.
-- Norbert Wiener
https://www.cse.buffalo.edu/~knepley/<https://urldefense.us/v3/__http:/www.cse.buffalo.edu/*knepley/__;fg!!G2kpM7uM-TzIFchu!nue-xIlrKIjtG6dGeWKiWVhSxLIOor_uLXP0UEel7pqB4YUy0y-YTHDqVX9IQCFFohVy9g$>
