<html><head><meta http-equiv="Content-Type" content="text/html; charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;" class=""><div class=""><br class=""></div><div class=""> u_t = G(u)</div><div class=""><br class=""></div> 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.<div class=""><div class=""><br class=""></div><div class=""> Barry</div><div class=""><br class=""><div><br class=""><blockquote type="cite" class=""><div class="">On Mar 22, 2021, at 7:20 PM, Matthew Knepley <<a href="mailto:knepley@gmail.com" class="">knepley@gmail.com</a>> wrote:</div><br class="Apple-interchange-newline"><div class=""><div dir="ltr" class=""><div dir="ltr" class="">On Mon, Mar 22, 2021 at 7:53 PM Salazar De Troya, Miguel via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov" class="">petsc-users@mcs.anl.gov</a>> wrote:<br class=""></div><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
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<div class="gmail-m_5864116693622373823WordSection1"><p class="MsoNormal"><span lang="ES" style="font-size:11pt" class="">Hello<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span lang="ES" style="font-size:11pt" class=""><u class=""></u> <u class=""></u></span></p><p class="MsoNormal"><span style="font-size:11pt" class="">I am interested in implementing the LDG method in “A local discontinuous Galerkin method for directly solving Hamilton–Jacobi equations”
<a href="https://www.sciencedirect.com/science/article/pii/S0021999110005255" target="_blank" class="">https://www.sciencedirect.com/science/article/pii/S0021999110005255</a>. The equation is more or less of the form (for 1D case):<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span style="font-size:11pt" class=""> </span><span lang="ES" style="font-size:11pt" class="">p1 = f(u_x)<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span lang="ES" style="font-size:11pt" class=""> p2 = g(u_x)<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span lang="ES" style="font-size:11pt" class=""> u_t = H(p1, p2)<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span lang="ES" style="font-size:11pt" class=""><u class=""></u> <u class=""></u></span></p><p class="MsoNormal"><span style="font-size:11pt" class="">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?</span></p></div></div></blockquote><div class=""><br class=""></div><div class="">Hi Miguel,</div><div class=""><br class=""></div><div class="">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</div><div class="">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</div><div class="">to the LHS and have Backward Euler, but it is my rule of thumb.</div><div class=""><br class=""></div><div class="">So, were you looking for an IMEX scheme? If so, which terms should be lagged? Also, from the equations above, it is hard to</div><div class="">see why you need a solve to calculate p1/p2. It looks like just a forward application of an operator.</div><div class=""><br class=""></div><div class=""> Thanks,</div><div class=""><br class=""></div><div class=""> Matt</div><div class=""> </div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div lang="EN-US" style="overflow-wrap: break-word;" class=""><div class="gmail-m_5864116693622373823WordSection1"><p class="MsoNormal"><span style="font-size:11pt" class="">I am trying to solve the Hamilton-Jacobi equation u_t – H(u_x) = 0. I welcome any suggestion for better methods.<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span style="font-size:11pt" class=""><u class=""></u> <u class=""></u></span></p><p class="MsoNormal"><span style="font-size:11pt" class="">Thanks<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span style="font-size:11pt" class="">Miguel<u class=""></u><u class=""></u></span></p><p class="MsoNormal"><span style="font-size:11pt" class=""><u class=""></u> <u class=""></u></span></p><p class="MsoNormal"><span lang="ES" style="font-size: 9pt; font-family: Consolas;" class="">Miguel A. Salazar de Troya</span><span lang="ES" style="font-size: 10.5pt;" class=""><u class=""></u><u class=""></u></span></p>
<div class=""><p class="MsoNormal"><span style="font-size: 9pt; font-family: Consolas;" class="">Postdoctoral Researcher, Lawrence Livermore National Laboratory</span><span style="font-size: 10.5pt;" class=""><u class=""></u><u class=""></u></span></p>
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</blockquote></div><br clear="all" class=""><div class=""><br class=""></div>-- <br class=""><div dir="ltr" class="gmail_signature"><div dir="ltr" class=""><div class=""><div dir="ltr" class=""><div class=""><div dir="ltr" class=""><div class="">What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br class="">-- Norbert Wiener</div><div class=""><br class=""></div><div class=""><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank" class="">https://www.cse.buffalo.edu/~knepley/</a><br class=""></div></div></div></div></div></div></div></div>
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