On the time-stepping stability of continuous mass-lumped and discontinuous Galerkin finite elements for the 3D acoustic wave equation

Abstract

We solve the three-dimensional acoustic wave equation, discretized on tetrahedral meshes. Two methods are considered: mass-lumped continuous finite elements and the symmetric interior-penalty discontinuous Galerkin method (SIP-DG). Combining the spatial discretization with the leap-frog time-stepping scheme, which is second-order accurate and conditionally stable, leads to a fully explicit scheme. We provide estimates of its stability limit for simple cases, namely, the reference element with Neumann boundary conditions, its distorted version of arbitrary shape, the unit cube that can be partitioned into 6 tetrahedra with periodic boundary conditions, and its distortions. The CFL stability limit contains a length scale for which we considered different options. The one based on the sum of the eigenvalues of the spatial operator for the first degree mass-lumped element gives the best results. It resembles the diameter of the inscribed sphere but is slightly easier to compute. The stability estimates show that mass-lumped continuous and SIP-DG finite elements have comparable stability conditions, with the exception of the elements of the first degree. The stability limit for the mass-lumped elements is less restrictive and allows for larger time steps.