Numerical Analysis and Scientific Computing Seminar | Florent Renac, Invariant domain preserving discontinuous spectral approximations of hyperbolic systems of conservation laws

For Zoom Link please contact ddelreyfernandez@uwaterloo.ca  

Speaker

Research Scientist Florent Renac, The French Aerospace Lab ONERA

Title

Invariant domain preserving discontinuous spectral approximations of hyperbolic systems of conservation laws

Abstract

Florent Renac (DAAA, ONERA, Université Paris Saclay F-92322 Châtillon, France)

Joint work with Valentin Carlier (Max-Planck Institute for plasma physics, Garching, Germany)

The numerical approximation of solutions to nonlinear hyperbolic conservation laws with discontinuous spectral methods offers important advantages such as high-order accuracy, parallel efficiency, flexibility, etc. These methods however suffer from spurious oscillations around discontinuities of the exact solution due to Gibbs phenomenon that may lead to robustness issues and a lot of research work addresses such problem.

In this talk, we consider the solution limiters from Zhang and Shu [1,2] based on two key ingredients: (i) a CFL condition on the time step to guarantee that the cell-averaged approximate solution is positive, (ii) a limiter that locally scales the high-order solution around its cell-average making it positive. Up to our knowledge, this approach is however roughly limited to meshes with linear elements (Cartesian and triangular meshes) and we propose [3] to generalize the CFL condition in (i) to:

 - polygonal and polyhedral meshes (with possibly curved cells);

 - conservative schemes preserving uniform states.

We will give details on the derivation of this CFL condition and show some numerical illustrations with RKDG and DGSEM schemes for the simulation of the compressible gas dynamics.

[1] X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229 (2010), pp. 3091–3120.

[2] X. Zhang and C.-W. Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229 (2010), pp. 8918–8934.

[3] V. Carlier and F. Renac, Invariant domain preserving high-order spectral discontinuous approximations of hyperbolic systems, to appear in SIAM J. Sci. Comput., arXiv:2203.05452 [math.NA], 2023.