Abstract
In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving the second-order wave equation. We show that it is possible to implement an interface scheme of “penalty” type for the second-order wave equation, similar to the ones used for first-order hyperbolic and parabolic equations, and the second-order scheme used by Mattsson et al. 2008. These schemes, known as SAT schemes for finite difference approximations and penalties for spectral ones, and ours share similar properties but in our case one needs to pass at the interface a smaller amount of data than previously known schemes. This is important for multi-block parallelizations in several dimensions, for it implies that one obtains the same solution quality while sharing among different computational grids only a fraction of the data one would need for a comparable (in accuracy) SAT or Mattsson et al.’s scheme. The semi-discrete approximation used here preserves the norm and uses standard finite-difference operators satisfying summation by parts. For the time integrator we use a semi-implicit IMEX Runge–Kutta method. This is crucial, since the explicit Runge–Kutta method would be impractical given the severe restrictions that arise from the stiff parts of the equations.
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Notes
Here we aim at an accuracy of about one part in \(10^{3}\) for \(10\) periods. Enough to keep the phase without appreciable error for about \(10\) crossing times.
This is of course true in the case of scalar quantities. In the case we were dealing with systems of wave equations applied to tensor quantities, some coordinate transformations are unavoidable at interfaces.
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Acknowledgments
We thank Luis Lehner and Manuel Tiglio for discussions, and SeCyT-UNC, CONICET, FONCyT and the Partner Group grant of the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) for financial support. O.R. thanks Perimeter Institute for hospitality, where part of this research was carried out.
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Parisi, F., Cécere, M., Iriondo, M. et al. Numerical Treatment of Interfaces for Second-Order Wave Equations. J Sci Comput 62, 875–897 (2015). https://doi.org/10.1007/s10915-014-9880-7
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DOI: https://doi.org/10.1007/s10915-014-9880-7