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<p class="MsoNormal">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:<o:p></o:p></p>
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<p class="MsoNormal"> F(U', U, t) = G(t,U) <o:p></o:p></p>
<p class="MsoNormal"> p1 = f(u_x)<o:p></o:p></p>
<p class="MsoNormal"> p2 = g(u_x)<o:p></o:p></p>
<p class="MsoNormal"> <span lang="ES">u' - H(p1, p2) = 0<o:p></o:p></span></p>
<p class="MsoNormal"><span lang="ES"><o:p> </o:p></span></p>
<p class="MsoNormal">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<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">dF/dU = [[M, 0, 0], <o:p></o:p></p>
<p class="MsoNormal"> [0, M, 0], <o:p></o:p></p>
<p class="MsoNormal"> [H'(p1), H'(p2), \sigma*M]]<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Thanks<o:p></o:p></p>
<p class="MsoNormal">Miguel<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">Barry Smith <bsmith@petsc.dev><br>
<b>Date: </b>Monday, March 22, 2021 at 7:42 PM<br>
<b>To: </b>Matthew Knepley <knepley@gmail.com><br>
<b>Cc: </b>"Salazar De Troya, Miguel" <salazardetro1@llnl.gov>, "Jorti, Zakariae via petsc-users" <petsc-users@mcs.anl.gov><br>
<b>Subject: </b>Re: [petsc-users] Local Discontinuous Galerkin with PETSc TS<o:p></o:p></span></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"> u_t = G(u)<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"> 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.<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"> Barry<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">On Mar 22, 2021, at 7:20 PM, Matthew Knepley <<a href="mailto:knepley@gmail.com">knepley@gmail.com</a>> wrote:<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">On Mon, Mar 22, 2021 at 7:53 PM Salazar De Troya, Miguel via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov">petsc-users@mcs.anl.gov</a>> wrote:<o:p></o:p></p>
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<span lang="ES">Hello</span><o:p></o:p></p>
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<span lang="ES"> </span><o:p></o:p></p>
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I am interested in implementing the LDG method in “A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations”
<a href="https://urldefense.us/v3/__https:/www.sciencedirect.com/science/article/pii/S0021999110005255__;!!G2kpM7uM-TzIFchu!nue-xIlrKIjtG6dGeWKiWVhSxLIOor_uLXP0UEel7pqB4YUy0y-YTHDqVX9IQCHtstz33g$" target="_blank">
https://www.sciencedirect.com/science/article/pii/S0021999110005255</a>. The equation is more or less of the form (for 1D case):<o:p></o:p></p>
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<span lang="ES">p1 = f(u_x)</span><o:p></o:p></p>
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<span lang="ES"> p2 = g(u_x)</span><o:p></o:p></p>
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<span lang="ES"> u_t = H(p1, p2)</span><o:p></o:p></p>
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<span lang="ES"> </span><o:p></o:p></p>
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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?<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">Hi Miguel,<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">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<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">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<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">to the LHS and have Backward Euler, but it is my rule of thumb.<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">So, were you looking for an IMEX scheme? If so, which terms should be lagged? Also, from the equations above, it is hard to<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">see why you need a solve to calculate p1/p2. It looks like just a forward application of an operator.<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"> Thanks,<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"> Matt<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"> <o:p></o:p></p>
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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.<o:p></o:p></p>
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<o:p></o:p></p>
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Thanks<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:.5in">
Miguel<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;margin-left:.5in">
<o:p></o:p></p>
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<span lang="ES" style="font-size:9.0pt;font-family:Consolas">Miguel A. Salazar de Troya</span><o:p></o:p></p>
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<span style="font-size:9.0pt;font-family:Consolas">Postdoctoral Researcher, Lawrence Livermore National Laboratory</span><o:p></o:p></p>
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<span style="font-size:9.0pt;font-family:Consolas">B141</span><o:p></o:p></p>
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<span style="font-size:9.0pt;font-family:Consolas">Rm: 1085-5</span><o:p></o:p></p>
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<span style="font-size:9.0pt;font-family:Consolas">Ph: 1(925) 422-6411</span><o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">-- <o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
-- Norbert Wiener<o:p></o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-left:.5in"><a href="https://urldefense.us/v3/__http:/www.cse.buffalo.edu/*knepley/__;fg!!G2kpM7uM-TzIFchu!nue-xIlrKIjtG6dGeWKiWVhSxLIOor_uLXP0UEel7pqB4YUy0y-YTHDqVX9IQCFFohVy9g$" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><o:p></o:p></p>
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