Collaborative Research: NSF-SNSF: Discontinuous Galerkin Methods on Cube-Sphere Mesh, Optimized for Long Time, Exascale, Structure-Preserving Simulations

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Interpreting many of the results from astronomical observatories requires numerical simulations, often run on supercomputers. These include dynamical radiative magnetohydrodynamic (MHD) simulations with algorithmic innovations, many of which are not currently in hand. Many of these innovations can be described as structure-preserving; in other words. there is physics in the numerical partial differential equations (PDEs) that has to be mimicked at a discrete level in codes that are optimized for spherical geometry. Our current capabilities for computational astrophysics are deficient because the relevant applied mathematics has not been developed for solving these problems. A research collaboration between the University of Notre Dame, Brown University and ETH Zürich in Switzerland will work together to overcome some of these deficiencies by making targeted advances in applied mathematics, which would be transformative in how they enable unprecedentedly novel astrophysical simulations that help in the interpretation of valuable observational data. The students and postdocs trained in this project will find many fertile career trajectories in astrophysics, applied math, and other fields such as plasma physics. The PI also runs an after-school remedial math program for students in the Gary, Indiana area who have fallen far behind in their math education. The applied mathematics challenges come in three parts: 1) We need high order divergence-preserving methods for capturing MHD turbulence that occurs in the vicinity of massive stellar winds. 2) We need multigroup radiation hydrodynamics 3) For long-term, high-fidelity, simulation of planetary atmospheres, we need well-balanced methods that are implicit in the radial direction and can preserve angular momentum. To overcome deficiencies in current astrophysical codes, the team will develop: 1) Discontinuous Galerkin (DG) methods with non-oscillatory design and a capability for preserving geometrical conservation laws (GCL) 2) Divergence-free DG methods that can operate on geometrically complex cube sphere meshes. 3) An efficient multi-group method for radiation hydrodynamics based on DG schemes. 4) Well-balanced DG schemes that are implicit in the radial direction so that very thin zones in the radial direction can be handled. 5) DG schemes that are angular momentum preserving. The MHD and radiation hydrodynamics innovations will also find engineering applications in fields as plasma-based space propulsion and in simulating the physics of tokamaks. This collaborative U.S.-Swiss project is supported by the U.S. National Science Foundation (NSF) and the Swiss National Science Foundation (SNSF), where NSF funds the U.S. investigator and SNSF funds the partners in Switzerland. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.