Summation-by-parts on staggered grids

Ossian O'Reilly
Department of Geophysics
Stanford University
Stanford, California, USA


High-order finite difference methods on staggered grids are widely used in computational seismology. These methods solve the elastic wave equation in first order form and possess excellent dispersion properties. However, they are mostly restricted to Cartesian geometries and uniform grid spacing.

In order to gain geometric flexibility and extend these methods to a larger class of problems, I present new summation-by-parts (SBP) operators on multiblock staggered grids. Unlike the traditional SBP operators, these SBP staggered grid operators rely on a modified summation-by-parts property and use a different quadrature rule on each grid. Boundary conditions can be weakly enforced using the Simultaneous-Approximation-Term (SAT) penalty technique. Alternately, by adopting a quadrature rule with zero weights at the endpoints, the boundary conditions can also be strongly enforced in some cases. Furthermore, these SBP operators can be applied to problems with coordinate singularities. Finally, I present some preliminary work and discussion of the challenges involved in extending the approach to curvilinear coordinates.