Energy Conserving Explicit Local Time-Stepping for Second-Order Wave Equations

Julien Diaz (1) , Marcus J. Grote (2)

1 MAGIQUE-3D - Magique 3D. Modélisation Avancée en GéophysIQUE 3D
2 Department of Mathematics [Basel]

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stability restriction, local time-stepping methods are developed, which allow arbitrarily small time-steps precisely where small elements in the mesh are located. When combined with a symmetric finite element discretization in space with an essentially diagonal mass matrix, the resulting discrete numerical scheme is explicit, inherently parallel, and exactly conserves a discrete energy. Starting from the standard second-order ``leap-frog'' scheme, time-stepping methods of arbitrary order of accuracy are derived. Numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory.

Mots clés: Second-order hyperbolic problems; explicit methods; time reversible methods; energy conservation; discontinuous Galerkin methods; finite element methods; mass lumping; wave equation; acoustic waves; electromagnetic waves; elastic waves

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

Julien Diaz, Marcus J. Grote. Energy Conserving Explicit Local Time-Stepping for Second-Order Wave Equations. [Research Report] 2007, pp.34. ⟨inria-00193160v1⟩