The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method

Abstract

The periodic standing wave (PSW) method for the binary inspiral of black holes and neutron stars computes exact numerical solutions for periodic standing wave spacetimes and then extracts approximate solutions of the physical problem, with outgoing waves. The method requires solution of a boundary value problem with a mixed (hyperbolic and elliptic) character. We present here a new numerical method for such problems, based on three innovations: (i) a coordinate system adapted to the geometry of the problem, (ii) an expansion in multipole moments of these coordinates and a filtering out of higher moments, and (iii) the replacement of the continuum multipole moments with their analogs for a discrete grid. We illustrate the efficiency and accuracy of this method with nonlinear scalar model problems. Finally, we take advantage of the ability of this method to handle highly nonlinear models to demonstrate that the outgoing approximations extracted from the standing wave solutions are highly accurate even in the presence of strong nonlinearities.

arXiv commentRevTex, 32 pages, 13 figures, 6 tables

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• Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains
• The periodic standing-wave approximation: post-Minkowski computation
• The periodic standing-wave approximation: computations in full general relativity
• Standing waves in general relativity
• Helical Symmetry in Linear Systems
• A fast stroboscopic spectral method for rotating systems in numerical relativity
• Numerical study of the Kerr solution in rotating coordinates