Editing Appendix/Ramblings/ForCohlHoward (section)

==Question Regarding Pressure-Poisson Equation==

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In an email dated 07 January 2023, Howard asked, "Do you think the pressure Poisson formulation of the Euler equations coupled with a 3D Poisson solver for gravity and evolution of the momentum equation might be our best strategy to evolve unstable Riemann S-type ellipsoids? We’ve already identified two different pieces of software (Fenics and COMSOL) which look like they might have this ability using finite-element. We also now have access to software which should be able to build good grids for FEM. Thanks! "
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Joel's response (11 January 2023) &hellip;

Your suggested "coupled Poisson formulation" sounds like a smart way to proceed.  Note, however, that I say this with limited authority; I have never written a code to solve the Euler equations that is based on a pressure-Poisson formulation.  My confidence in offering support for this approach comes from the following historical note.


In the early 1990s, in collaboration with Priya Vashishta, I helped the LSU Physics & Astronomy department secure approximately $0.8 million from the Louisiana Board of Regents to purchase [[Appendix/Ramblings/MyDoctoralStudents#Years_1988_-_1994|an 8K-node SIMD architecture MasPar MP1]].  As you know, our astrophysics group developed a 3D Poisson solver that was extraordinally well suited to the 8K-node hardware layout of the MasPar.  Around the same time that we received this ''state'' funding, I heard that the U.S. Department of Energy (DOE) was expecting to fund a limited number of large-scale fluid-flow (and heat-transfer) simulations in support of the aircraft industry's efforts to design better turbine engines.  LSU's research office organized a meeting of (primarily engineering) faculty to see if a group could be put together to submit a viable research proposal to the DOE.  I attended this meeting, just for grins; but it turned out to be a very smart decision.  At the meeting, I met [https://www.lsu.edu/eng/mie/people/formerfaculty/acharya.php Sumanta Acharya] (and [https://www.lsu.edu/eng/mie/people/faculty/nikitopoulos.php Dimitris Nikitopoulos]) and learned about his multidimensional simulations of incompressible fluids; he was looking for funding opportunities that would support his efforts to scale from 2D to 3D simulations.  I also learned that he usually solved the Euler equations using a pressure-Poisson approach.  We both realized that my group's successful implementation of a 3D Poisson solver (for gravity) on the MasPar could serve as an illustration of how his group could likely efficiently solve the pressure-Poisson equation in 3D.  We wrote the proposal to DOE, and it was funded at $0.5 million.  Thus began a very successful, multi-decade collaboration between my group and Sumanta's mechanical engineering group.


So &hellip; I am fairly confident that the approach you take to solve the gravitational-Poisson equation will help you understand how to solve the pressure-Poisson equation (or vise versa) on the finite-element grid.  Note, however, that to my knowledge, Sumanta's simulations always involved a finite-difference, rather than a finite-element, methodology.