Runge--Kutta-Based Explicit Local Time-Stepping Methods for Wave Propagation

Abstract

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

Newmark local time stepping on high-performance computing architectures

“…Following the framework in [7,14,15], we divide the finite-element mesh into both fine and coarse element regions and correspondingly we split the DOFs as…”

Discontinuous Galerkin Time-Domain Modeling of Graphene Nanoribbon Incorporating the Spatial Dispersion Effects

“…To solve the semi-discrete matrix equations from (25) to (27) and together with (33) and (34), the explicit fourstage Runge-Kutta (RK) time-marching method [32], [33] is employed. To ensure stability, the maximum time step size δt is determined in terms of the following condition [13]:…”

Error Analysis of a Second-Order Locally Implicit Method for Linear Maxwell's Equations

“…Lemma 5.1. Let the exact solution satisfy u 2 C 0, T ; H k+1 (T h ) 6 . Under the CFL condition (4.1) the error e n h defined in (5.2) satisfies…”