Absorbing boundary conditions for simulation of gravitational waves with spectral methods in spherical coordinates

TL;DR

This paper introduces a new spectral boundary condition for simulating gravitational waves that effectively absorbs quadrupolar wave energy, improving accuracy in numerical relativity simulations.

Contribution

A novel multipolar boundary condition formulation using spectral methods, optimized for quadrupolar gravitational wave simulations in spherical coordinates.

Findings

Achieves high accuracy in wave absorption

Performs efficiently for dipolar and quadrupolar waves

Comparable to traditional Sommerfeld boundary conditions for monopolar waves

Abstract

We present a new formulation of the multipolar expansion of an exact boundary condition for the wave equation, which is truncated at the quadrupolar order. Using an auxiliary function, that is the solution of a wave equation on the sphere defining the outer boundary of the numerical grid, the absorbing boundary condition is simply written as a perturbation of the usual Sommerfeld radiation boundary condition. It is very easily implemented using spectral methods in spherical coordinates. Numerical tests of the method show that very good accuracy can be achieved and that this boundary condition has the same efficiency for dipolar and quadrupolar waves as the usual Sommerfeld boundary condition for monopolar ones. This is of particular importance for the simulation of gravitational waves, which have dominant quadrupolar terms, in General Relativity.