High-order uniformly accurate time integrators for semilinear wave equations of Klein-Gordon type in the non-relativistic limit

TL;DR

This paper presents high-order time integrators for Klein-Gordon semilinear wave equations that remain accurate as the speed of light increases, using a general oscillatory quadrature rule without needing nonlinearity-specific pre-computations.

Contribution

The authors develop a family of high-order, uniformly accurate time semi-discretizations for Klein-Gordon wave equations that do not require nonlinearity-specific pre-computations or step size restrictions.

Findings

Achieves uniform accuracy in the non-relativistic limit

Does not require pre-computations specific to the nonlinearity

No restrictions on step size

Abstract

We introduce a family of high-order time semi-discretizations for semilinear wave equations of Klein--Gordon type with arbitrary smooth nonlinerities that are uniformly accurate in the non-relativistic limit where the speed of light goes to infinity. Our schemes do not require pre-computations that are specific to the nonlinearity and have no restrictions in step size. Instead, they rely upon a general oscillatory quadrature rule developed in a previous paper (Mohamad and Oliver, arXiv:1909.04616).