Strongly consistent marching schemes for the wave equation

Authors:
Jing-Rebecca Li;Leslie Greengard
Affiliations:
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY;Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY
Venue:
Journal of Computational Physics
Year:
2003

Citing 4
Cited 3

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Finite difference schemes and partial differential equations
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High order marching schemes for the wave equation in complex geometry

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Abstract

In this paper, we consider a class of explicit marching schemes first proposed in [1] for solving the wave equation in complex geometry using an embedded Cartesian grid. These schemes rely on an integral evolution formula for which the numerical domain of dependence adjusts automatically to contain the true domain of dependence. Here, we refine and analyze a subclass of such schemes, which satisfy a condition we refer to as strong u-consistency. This requires that the evolution scheme be exact for a single-valued approximation to the solution at the previous time steps. We provide evidence that many of these strongly u-consistent schemes are stable and converge at very high order even in the presence of small cells in the grid, while taking time steps dictated by the uniform grid spacing.