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 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, 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 accuracy are derived. Numerical experiments illustrate the efficiency and usefulness of these methods and validate the theory. Introduction.The efficient and accurate numerical solution of the wave equation is of fundamental importance for the simulation of time dependent acoustic, electromagnetic, or elastic wave phenomena. Finite difference methods are commonly used for the simulation of time dependent waves because of their simplicity and their efficiency on structured Cartesian meshes [27,28,34]. However, in the presence of complex geometry or small geometric features that require locally refined meshes, their usefulness is somewhat limited. In contrast, finite element methods (FEMs) easily handle locally refined unstructured meshes; moreover, their extension to high order is straightforward, even in the presence of curved boundaries or material interfaces.The finite element Galerkin discretization of second-order hyperbolic problems typically leads to a second-order system of ordinary differential equations. Even if explicit time stepping is employed, the mass matrix arising from the spatial discretization by standard continuous finite elements must be inverted at each time step-a major drawback in terms of efficiency. To overcome that problem, various "mass lumping" techniques have been proposed, which effectively replace the mass matrix by a diagonal approximation. While straightforward for piecewise linear elements [8,32], mass lumping techniques require particular quadrature or cubature rules at higher order to preserve the accuracy and guarantee numerical stability [14,22].Alternatively, discontinuous Galerkin (DG) methods offer even greater flexibility for local mesh refinement by accommodating nonconforming grids and hanging nodes.

Component splitting for semi-discrete Maxwell equations

“…1, the method from [2] and [4] also serves to overcome step size limitations by grid-induced stiffness. That method is designed for second-order wave equations and the work reported focuses also on the Maxwell equations.…”

“…The new scheme also bears a relationship to a component splitting scheme discussed in [3] which is especially designed for a discontinuous Galerkin discretization. A novel local time-stepping technique for second-order wave equations discretized in space by a continuous or discontinuous finite element method has been proposed in [2] and is further discussed in [4]. This local time-stepping technique also serves to overcome step size limitations by grid-induced stiffness.…”

“…5 on some plans for future work. Also a few remarks on the approach from [2] and [4] are included in this final section.…”

Multi-level explicit local time-stepping methods for second-order wave equations

“…By combining a symplectic integrator with a DG discretization of Maxwell's equations in first-order form, Piperno [18] proposed a second-order explicit local time-stepping scheme, which also conserves a discrete energy. Starting from the standard LF method, the authors proposed energy conserving fully explicit LTS integrators of arbitrarily high accuracy for the wave equation [1]; that approach was extended to Maxwell's equations in [19]. An hp-version, where not only the time-step but also the order of approximation is adapted within di↵erent regions of the mesh, was proposed in [20] and later applied to a realistic geological model [21].…”

“…Starting from a semi-discrete Galerkin finite element formulation of the wave equation, we derive in Section 2 local time-stepping (LTS) methods of arbitrarily high order based on the leap-frog (LF) method; we also recall some of their key properties from [1]. Although first presented in [1], the present derivation is di↵erent and crucial for the derivation of the multi-level local time-stepping (MLTS) methods in Section 3. In Section 4, we prove that the second-order MLTS method conserves a discrete energy regardless of the number of intermediate levels.…”