Locally refined meshes severely impede the efficiency of explicit Runge-Kutta (RK) methods for the simulation of time-dependent wave phenomena. By taking smaller time-steps precisely where the smallest elements are located, local time-stepping (LTS) methods overcome the bottleneck caused by the stringent stability constraint of but a few small elements in the mesh. Starting from classical or low-storage explicit RK methods, explicit LTS methods of arbitrarily high accuracy are derived. When combined with an essentially diagonal finite element mass matrix, the resulting time-marching schemes retain the high accuracy, stability, and efficiency of the original RK methods while circumventing the geometry-induced stiffness. Numerical experiments with continuous and discontinuous Galerkin finite element discretizations corroborate the expected rates of convergence and illustrate the usefulness of these LTS-RK methods.1. Introduction. The efficient numerical simulation of time-dependent wave phenomena is of fundamental importance in acoustic, electromagnetic, or seismic wave propagation. In the presence of heterogeneous media or complex geometry, finite element methods (FEM), be they continuous or discontinuous, are increasingly popular due to their inherent flexibility: elements can be small precisely where small features are located and larger elsewhere. Local mesh refinement, however, also imposes severe stability constraints on explicit time integration, as the maximal time-step is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small region, the use of implicit methods, or a very small time-step in the entire computational domain, are generally too high a price to pay. Local time-stepping (LTS) methods alleviate that geometry, induced stability restriction by dividing the elements into two distinct regions: the "coarse region," which contains the larger elements and is integrated in time using an explicit method, and the "fine region," which contains the smaller elements and is integrated in time using either smaller time-steps or an implicit scheme.Locally implicit methods build on the long tradition of hybrid implicit-explicit (IMEX) algorithms for operator splitting in computational fluid dynamics-see [36,2] and the references therein. In 2006, Piperno [37] combined the explicit leap-frog (LF) with the implicit Crank-Nicolson (CN) scheme for a nodal discontinuous Galerkin (DG) discretization of Maxwell's equations in a nonconducting medium. Here, a linear system needs to be solved inside the refined region at every time-step. Although each method is time accurate of order two, the implicit-explicit component splitting

Local adaptivity and mesh refinement are key to the efficient simulation of wave phenomena in heterogeneous media or complex geometry. Locally refined meshes, however, dictate a small time-step everywhere with a crippling effect on any explicit time-marching method. In [18] a leap-frog (LF) based explicit local time-stepping (LTS) method was proposed, which overcomes the severe bottleneck due to a few small elements by taking small time-steps in the locally refined region and larger steps elsewhere. Here a rigorous convergence proof is presented for the fullydiscrete LTS-LF method when combined with a standard conforming finite element method (FEM) in space. Numerical results further illustrate the usefulness of the LTS-LF Galerkin FEM in the presence of corner singularities.

Local mesh refinement significantly influences the performance of explicit timestepping methods for numerical wave propagation. Local time-stepping (LTS) methods improve the efficiency by using smaller time-steps precisely where the smallest mesh elements are located, thus permitting a larger time-step in the coarser regions of the mesh without violating the stability condition. However, when the mesh contains nested patches of refinement, any local time-step will be unnecessarily small in some regions. To allow for an appropriate time-step at each level of mesh refinement, multi-level local time-stepping (MLTS) methods have been proposed. Starting from the Runge-Kutta-based LTS methods derived by Grote et al. [17], we propose explicit MLTS methods of arbitrarily high accuracy. Numerical experiments with finite difference and continuous finite element spatial discretizations illustrate the usefulness of the novel MLTS methods and show that they retain the high accuracy and stability of the underlying Runge-Kutta methods.

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