On high-precision L^∞-stable IMEX schemes for scalar hyperbolic multi-scale equations

Victor Michel-Dansac (1, 2, 3) , Andrea Thomann (4, 5)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse
3 TONUS - TOkamaks and NUmerical Simulations
4 Dipartimento di Scienze e Alta Tecnologia
5 Institut für Mathematik [Mainz]

We present a framework to build high-precision IMEX schemes that fulfill the maximum principle, applied to a scalar hyperbolic multi-scale equation. Motivated by the findings in [5] that implicit R-K schemes are not L∞ stable, our scheme, for which we can prove the L∞ stability, is based on a convex combination between a first-and a class of second-order IMEX schemes. We numerically demonstrate the advantages of our scheme, especially for discontinuous problems, and give a MOOD procedure to increase the precision.

Mots clés: L$∞$ stability; IMEX R-K schemes; MOOD; hyperbolic multi-scale equations

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Victor Michel-Dansac, Andrea Thomann. On high-precision L^∞-stable IMEX schemes for scalar hyperbolic multi-scale equations. Recent Advances in Numerical methods for Hyperbolic PDE Systems. Selected talks of Numhyp 2019, Jun 2019, Málaga, Spain. ⟨10.1007/978-3-030-72850-2_4⟩. ⟨hal-02303491v3⟩