A New Way to Make Waves

TL;DR

This paper introduces a novel algorithm for solving nonlinear wave equations that operates on characteristic hypersurfaces, improving accuracy and efficiency in simulating wave phenomena, especially in gravitational wave modeling.

Contribution

The paper presents a new characteristic hypersurface-based algorithm applicable to various hyperbolic systems, enhancing simulation accuracy and efficiency for nonlinear waves and gravitational signals.

Findings

More accurate and efficient than existing methods for Cauchy boundary conditions

Successfully applied to simulate gravitational waves from black hole collisions

Demonstrates potential for broad application to hyperbolic PDEs

Abstract

I describe a new algorithm for solving nonlinear wave equations. In this approach, evolution takes place on characteristic hypersurfaces. The algorithm is directly applicable to electromagnetic, Yang-Mills and gravitational fields and other systems described by second differential order hyperbolic equations. The basic ideas should also be applicable to hydrodynamics. It is an especially accurate and efficient way for simulating waves in regions where the characteristics are well behaved. A prime application of the algorithm is to Cauchy-characteristic matching, in which this new approach is matched to a standard Cauchy evolution to obtain a global solution. In a model problem of a nonlinear wave, this proves to be more accurate and efficient than any other present method of assigning Cauchy outer boundary conditions. The approach was developed to compute the gravitational wave signal produced by collisions of two black holes. An application to colliding black holes is presented.