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Séminaire du LJLL - 08 01 2021 14h00 : A. Bressan
08 janvier 2021 — 14h00
Exposé à distance retransmis par Zoom
Alberto Bressan (Université d’Etat de Pennsylvanie)
A posteriori error estimates for numerical solutions to hyperbolic conservation laws
Résumé
For general n x n hyperbolic systems of conservation laws in one space dimension, several approximation methods are known to converge to the unique entropy admissible solution : the Glimm scheme, front tracking, vanishing viscosity, semidiscrete approximations, and second order hyperbolic regularizations. In some cases, a priori error estimates and asymptotic convergence rates are also available. Notably absent from this list are all the fully discrete numerical schemes, such as the Godunov or the Lax-Friedrichs scheme, where the convergence to a unique limit remains an open problem.
After explaining the key obstruction toward a derivation of a priori estimates for these schemes, this talk will focus on a posteriori error estimates. The main issue to be discussed is the following : given a numerically computed approximation, what do we need to check, in order to conclude that the discrete approximation is close to the exact solution ?
Some results in this direction will be presented, including a "post-processing algorithm" and numerical simulations.
This is a joint work with Maria Teresa Chiri and Wen Shen.