Energy conserving explicit local time-stepping for second-order wave equations

Abstract. Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical simulation of time dependent wave phenomena. To overcome that stabil-ity 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 nu-merical scheme is explicit, is inherently parallel, and exactly conserves a discrete energy. Starting from the standard second-order “leap-frog ” scheme, time-stepping methods of arbitrary order of ac-curacy are derived. Numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory. Key words. second-order hyperbolic problems, explicit methods, time reversible methods, energy conservation, discontinuous Galerkin methods, finite element methods, mass lumping, wave equation, acoustic waves, electromagnetic wave