Relaxing the CFL condition for the wave equation on adaptive meshes

Abstract

The Courant-Friedrichs-Lewy (CFL) condition guarantees the stability of the popular explicit leapfrog method for the wave equation. However, it limits the choice of the time step size to be bounded by the minimal mesh size in the spatial finite element mesh. This essentially prohibits any sort of adaptive mesh refinement that would be required to reveal optimal convergence rates on domains with re-entrant corners. A simple subspace projection step inspired by numerical homogenisation can remove the critical time step restriction so that the CFL condition and approximation properties are balanced in an optimal way, even in the presence of spatial singularities. We consider the discretisation of the wave equation, with (possibly) re-entrant corners. This typically reduces the regularity of the solution and leads to u(t) ∈ H 2 (Ω). To reveal optimal convergence rates, non-uniform mesh refinement in space proves advantageous for the wave equation [2]. Given a discrete space V H , the leapfrog scheme seeks a solution (u n H ) n=0,...,N with u n H ∈ V H such that for all n = 2, . . . , N and allAs usual for explicit time discretisation schemes, the numerical stability is conditional and guaranteed only under the sharp Courant-Friedrichs-Lewy (CFL) condition. In the present context of linear finite elements, it bounds the time step size by the minimal mesh size of the spatial mesh ∆t h min .While on (quasi-)uniform meshes the admissible choice ∆t ≈ h min ≈ h max is considered as a natural balance of space and time discretisation, the CFL condition is not at all compatible with non quasi-uniform meshes in the sense that the efficiency of adaptive mesh refinement in space causes tiny time steps that destroy the overall complexity. Essentially, the CFL condition forbids any type of spatial adaptivity.It is proved in [3] that the restriction of the time step by the minimal spatial mesh size can easily be removed by projecting the adaptive finite element space to some subspace V H with similar (optimal) approximation properties for weak solutions of the wave equation. The underlying technique is well-established in the context of numerical homogenisation [1]. The reduced space V H allows for an improved inverse inequality that decouples the time step from the minimal mesh size and turns the leapfrog into a feasible numerical scheme also on adaptive spatial meshes. Spatial reductionWe consider a quasi-uniform shape regular triangulation T H with (maximal) mesh size H and some (possibly) non-quasiuniform shape regular triangulation and refinement T h of T H with corresponding standard P 1 -FEM spaces S 1 0 (T H ) and S 1 0 (T h ). We assume that T h leads to an optimal approximation property in the sense thatThe construction of the generalised finite element space is based on a projective quasi-interpolation operator I H : V → S 1 0 (T H ) with approximation and stability properties

Explicit computational wave propagation in micro-heterogeneous media

“…The approach is based on the so-called Localized Orthogonal Decomposition method (LOD) introduced in [22] (see also [27]) and uses ideas similar to the ones presented in [29] for the wave equation in homogeneous media posed on domains with re-entrant corners. The basic idea of the method is to define lowdimensional finite element spaces that include spatial fine scale features.…”

Multiscale Methods for Wave Problems in Heterogeneous Media

“…Recently it has been shown that Approach 4 also intrinsically relaxes the CFL condition on adaptive meshes [47], which is very significant for corresponding time-discretizations. Generalizations of the approach to the Helmholtz equation in the context of high frequency wave propagation are given in [16,28,46].…”

Space-time discontinuous Galerkin approximation of acoustic waves with point singularities

“…One remedy with a space-time discretization flavour is here the so-called local time-stepping. We refer to Diaz-Grote [11] for details and to [40] for a method-of-lines based, related approach to circumvent the CFL-constraint in explicit time-marching.…”