Juliette Chabassier | Local time discretisation for linear wave equations

This page describes contributions that provide coupling of different time integration strategies, leading to local time integration. A domain decomposition approach allows to derive

a coupling of different fourth order schemes over a boundary,
a coupling of explicit and implicit second order scheme,
a coupling of explicit schemes with different stability conditions,
the space time convergence proof of the previous strategies, for a small enough time step

References

IJNME

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

Chabassier, Juliette, and Imperiale, Sébastien

International Journal for Numerical Methods in Engineering Mar 2015

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A family of fourth-order coupled implicit–explicit time 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 fourth-order time discretizations are used, chosen among a family of implicit or explicit fourth-order schemes. The coupling is based on a Lagrangian formulation on the boundaries between the several non-conforming meshes of the regions. A global discrete energy is shown to be preserved and leads to global fourth-order consistency in time. Numerical results in 1D and 2D for the acoustic and elastodynamics equations 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. Accuracy up to fourth order reduces the numerical dispersion inherent to implicit methods used with a large time step and makes this family of schemes attractive compared with second-order accurate methods.
M2AN

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

Chabassier, Juliette, and Imperiale, Sébastien

ESAIM: Mathematical Modelling and Numerical Analysis Jul 2021

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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.