Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

Juliette Chabassier (1, 2) , Sébastien Imperiale (3, 4)

1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
2 MAKUTU - Modélisation et simulation de la propagation des ondes fondées sur des mesures expérimentales pour caractériser des milieux géophysiques et héliophysiques et concevoir des objets complexes
3 M3DISIM - Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine
4 Université Paris-Saclay

In this work we present and analyse a time discretisation strategy for linear wave equations t hat aims at using locally in space the most adapted time discretisation among a family of implicit or explicit centered second order schemes. The proposed family of schemes is adapted to domain decomposition methods such as the mortar element method. They correspond in that case to local implicit schemes and to local time stepping. We show that, if some regularity properties of the solution are satisfied and if the time step verifies a stability condition, then the family of proposed time discretisations provides, in a strong norm, second order space-time convergence. Finally, we provide 1D and 2D numerical illustrations that confirm the obtained theoretical results and we compare our approach on 1D test cases to other existing local time stepping strategies for wave equations.

Mots clés: Local time stepping; Local implicit scheme; Converge analysis; Time discretisation; Wave equations

Domaines: Mathématiques [math]

Juliette Chabassier, Sébastien Imperiale. Construction and convergence analysis of conservative second order local time discretisation for linear wave equations. ESAIM: Mathematical Modelling and Numerical Analysis, 2021, 55 (4), pp.1507-1543. ⟨10.1051/m2an/2021030⟩. ⟨hal-03309010⟩