Convergence Analysis of Energy Conserving Explicit Local Time-Stepping Methods for the Wave Equation

Abstract

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.

Tent pitching and Trefftz-DG method for the acoustic wave equation

“…Classical time-stepping schemes can still be applied successfully by splitting the domain into a coarse-mesh and a fine-mesh region, then explicit time stepping in the coarse-mesh region is combined with local implicit or explicit time stepping in the fine-mesh region. A fully explicit scheme can be found in [12,13].…”

Construction and convergence analysis of conservative second order local time discretisation for linear wave equations

“…The price to pay is that a (hopefully small) linear problem must be solved at each iteration. o Local Time Stepping (LTS), see for instance [7][8][9][10][11]. The strategy is to use a first time marching scheme in the whole domain and a second one in the perturbed region.…”

“…It has first been proposed in [8] and various extensions exist: Maxwell's equations (see [30]) and multi-level LTS (see [31]). Recently, in [7] a proof of space-time convergence is given. It shows that, for the scalar wave equation, a second order space-time convergence holds in the L 2 norm in space.…”

A High-Order, Conservative Integrator with Local Time-Stepping

“…More recently, many methods have been investigated as starting points for more sophisticated LTS methods, including both substep [5,6,7,8,9,10] and multistep [2,11,12] integrators and also less common methods such as leapfrog [11,13], Richardson extrapolation [14], ADER [15], and implicit methods [16]. Demirel et al [17] have even explored LTS schemes constructed from multiple unrelated GTS integrators.…”

“…LTS integrators for the special case of linear systems have been developed based on Adams-Bashforth [11], Runge-Kutta [7,10], and leapfrog [11,13] schemes. Of particular interest here, starting from the Adams-Bashforth methods, Grote and Mitkova [11] found a family of high-order, conservative methods for integral ratios between step sizes on different degrees of freedom.…”