Fourth-order energy-preserving locally implicit time discretization for linear wave equations

Juliette Chabassier (1, 2) , Sebastien Imperiale (3)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]
2 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
3 M3DISIM - Mathematical and Mechanical Modeling with Data Interaction in Simulations for Medicine

A family of fourth order locally implicit schemes is presented as a special case of fourth order coupled implicit schemes for linear wave equations. The domain of interest is decomposed into several regions where different (explicit or implicit) fourth order time discretization are used. The coupling is based on a Lagrangian formulation on the boundaries between the several non conforming meshes of the regions. Fourth order accuracy follows from global energy identities. Numerical results in 1d and 2d illustrate the good behavior of the schemes and their potential for the simulation of realistic highly heterogeneous media or strongly refined geometries, for which using everywhere an explicit scheme can be extremely penalizing. Fourth order accuracy reduces the numerical dispersion inherent to implicit methods used with a large time step, and makes this family of schemes attractive compared to classical approaches.

Keywords: Time discretization; Locally implicit schemes; High-order numerical methods

Domains: Numerical Analysis [cs.NA]

Juliette Chabassier, Sebastien Imperiale. Fourth-order energy-preserving locally implicit time discretization for linear wave equations. WAVES 2015 - The 12th international conference on mathematical and numerical aspects of wave propagation, Jul 2015, Karlsruhe, Germany. ⟨10.1002/nme.5130⟩. ⟨hal-01264044⟩