A hyperboloidal method for numerical simulations of multidimensional nonlinear wave equations: nonlinear tails

TL;DR

This paper introduces a hyperboloidal numerical method to study the late-time decay of solutions to multidimensional nonlinear wave equations without symmetry assumptions, providing new insights into tail behaviors across various regimes.

Contribution

It extends numerical analysis of nonlinear wave tails beyond radial symmetry using hyperboloidal foliation and conformal compactification in multiple dimensions.

Findings

Decay exponents for different modes and regimes computed

Analysis includes subcritical, critical, and supercritical cases

Method applicable to n=3 and higher dimensions

Abstract

We consider the scalar wave equation with power nonlinearity in n+1 dimensions. Unlike most previous numerical studies, we go beyond the radial case and do not assume any symmetries for n=3, and we only impose an SO(n-1) symmetry in higher dimensions. Our method is based on a hyperboloidal foliation of Minkowski spacetime and conformal compactification. We focus on the late-time power-law decay (tails) of the solutions and compute decay exponents for different spherical harmonic modes, for subcritical, critical and supercritical, focusing and defocusing nonlinear wave equations.