Time-stepping stability of continuous and discontinuous finite-element methods for 3-D wave propagation

Time-stepping stability of continuous and discontinuous finite-element methods for 3-D wave propagation

Mulder, W.A.
Zhebel, E.
Minisini, S.

Civil Engineering and Geosciences

Geoscience & Engineering

2013-12-03

We analyse the time-stepping stability for the 3-D acoustic wave equation, discretized on tetrahedral meshes. Two types of methods are considered: mass-lumped continuous finite elements and the symmetric interior-penalty discontinuous Galerkin method. 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 six tetrahedra with periodic boundary conditions and its distortions. The Courant–Friedrichs–Lewy stability limit contains an element diameter 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 the mass-lumped continuous and the discontinuous Galerkin finite elements of degree 2 have comparable stability conditions, whereas the masslumped elements of degree one and three allow for larger time steps.

numerical solutions
Fourier analysis
numerical approximations and analysis
computational seismology
wave propagation

http://resolver.tudelft.nl/uuid:c31ed986-abde-484b-91a3-dad26d852716

Oxford University Press

0956-540X

https://doi.org/10.1093/gji/ggt446

Geophysical Journal International, 196 (2), 2014

Institutional Repository

journal article

(c) 2013 The Author(s)