Provably stable and high-order accurate finite difference schemes on staggered grids and their potential application for seismic hazard analysis

Ossian O'Reilly, & Eric M. Dunham

Published August 15, 2016, SCEC Contribution #6953, 2016 SCEC Annual Meeting Poster #320

When it comes to wave propagation over large distances, high-order and centered finite difference approximations on staggered grids are highly effective. For this reason, the 4th order staggered grid scheme is used for many of SCEC’s high performance computing efforts, ranging from large-scale earthquake hazard simulations like ShakeOut to full-waveform tomography to reciprocity-based CyberShake calculations. In these simulations, it is essential to accurately model surface waves due to their dominant role in ground motion at all but the closest distances. However, unless care is taken in the implementation of the free surface boundary condition, instability or inaccuracy can result. It is common to decrease the order of accuracy near the free surface or to introduce ad-hoc artificial dissipation or filtering to avoid instability. Unfortunately, these techniques can severely degrade the accuracy of the solution, reduce the maximum stable time step, or involve additional parameters that can be inconvenient for practitioners to tune. In this work, we construct a provably stable numerical scheme on staggered grids that retains high accuracy near boundaries. This is done by deriving one-sided finite difference approximations near the boundary that satisfy the principle of summation-by-parts (SBP). To preserve the SBP property of the scheme, and establish stability using energy estimates for the semi-discrete problem, boundary conditions are weakly enforced. Here we demonstrate the technique for the acoustic wave equation in Cartesian coordinates, but its extension to the elastic wave equation is anticipated to be straightforward. To handle complex geometry, the technique can likely be extended to curvilinear grids, non-conforming grids, or be coupled to other provably stable schemes such as discontinuous Galerkin schemes.

Citation
O'Reilly, O., & Dunham, E. M. (2016, 08). Provably stable and high-order accurate finite difference schemes on staggered grids and their potential application for seismic hazard analysis. Poster Presentation at 2016 SCEC Annual Meeting.

Related Projects & Working Groups
Earthquake Simulators