Research Project: Robust and Accurate Approximation of Hyperbolic Systems
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- Popov, Bojan
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Abstract or Project Summary
The project consists of developing robust numerical methods for solving
hyperbolic systems of conservation laws such as the Euler equations, the shallow water equations, and
related systems such as radiative hydrodynamics. Robustness is an area of investigation that is not well
addressed in the literature. We think that most current high-order methods are not robust. This lack
of robustness is what makes them unattractive to practitioners. Real progresses will come when robust
accurate methods are readily available to a large community of practitioners. We do not think that this
is the case at the moment. Our research program is articulated around the following five points: (i)
The numerical method must be invariant domain preserving (i.e. preserve convex invariant domains
of the problem) on any unstructured meshes in any space dimension; (ii) The method must be robust
in the sense that it should not involve any tuning parameter, mesh-dependent coefficient, or problemdependent
stabilization. A robust method should also be easy to program and parallelize. It should
not require any subtle mathematical knowledge from practitioners to run it; (iii) The method must be
at least third-order accurate in space and time and must be open to higher-order extensions; (iv) The
method should respect the physical dissipation of the PDE; (v) The above objectives must be reached by
stating precise statements supported either by mathematical proofs or very strong numerical evidences
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