<div dir="ltr"><div dir="ltr">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:<br></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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<p class="MsoNormal"><span lang="ES" style="font-size:11pt">Hello<u></u><u></u></span></p>
<p class="MsoNormal"><span lang="ES" style="font-size:11pt"><u></u> <u></u></span></p>
<p class="MsoNormal"><span style="font-size:11pt">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">https://www.sciencedirect.com/science/article/pii/S0021999110005255</a>. The equation is more or less of the form (for 1D case):<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11pt"> </span><span lang="ES" style="font-size:11pt">p1 = f(u_x)<u></u><u></u></span></p>
<p class="MsoNormal"><span lang="ES" style="font-size:11pt"> p2 = g(u_x)<u></u><u></u></span></p>
<p class="MsoNormal"><span lang="ES" style="font-size:11pt"> u_t = H(p1, p2)<u></u><u></u></span></p>
<p class="MsoNormal"><span lang="ES" style="font-size:11pt"><u></u> <u></u></span></p>
<p class="MsoNormal"><span style="font-size:11pt">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><br></div><div>Hi Miguel,</div><div><br></div><div>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>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>to the LHS and have Backward Euler, but it is my rule of thumb.</div><div><br></div><div>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>see why you need a solve to calculate p1/p2. It looks like just a forward application of an operator.</div><div><br></div><div> Thanks,</div><div><br></div><div> Matt</div><div> </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;"><div class="gmail-m_5864116693622373823WordSection1">
<p class="MsoNormal"><span style="font-size:11pt">I am trying to solve the Hamilton-Jacobi equation u_t – H(u_x) = 0. I welcome any suggestion for better methods.<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11pt"><u></u> <u></u></span></p>
<p class="MsoNormal"><span style="font-size:11pt">Thanks<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11pt">Miguel<u></u><u></u></span></p>
<p class="MsoNormal"><span style="font-size:11pt"><u></u> <u></u></span></p>
<p class="MsoNormal"><span lang="ES" style="font-size:9pt;font-family:Consolas;color:black">Miguel A. Salazar de Troya</span><span lang="ES" style="font-size:10.5pt;color:black"><u></u><u></u></span></p>
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<p class="MsoNormal"><span style="font-size:9pt;font-family:Consolas;color:black">Postdoctoral Researcher, Lawrence Livermore National Laboratory</span><span style="font-size:10.5pt;color:black"><u></u><u></u></span></p>
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<p class="MsoNormal"><span style="font-size:9pt;font-family:Consolas;color:black">B141</span><span style="font-size:10.5pt;color:black"><u></u><u></u></span></p>
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<p class="MsoNormal"><span style="font-size:9pt;font-family:Consolas;color:black">Rm: 1085-5</span><span style="font-size:10.5pt;color:black"><u></u><u></u></span></p>
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<p class="MsoNormal"><span style="font-size:9pt;font-family:Consolas;color:black">Ph: 1(925) 422-6411</span><u></u><u></u></p>
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</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr" class="gmail_signature"><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><div>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</div><div><br></div><div><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><br></div></div></div></div></div></div></div></div>