Error analysis of one-stage explicit extended RKN integrators for semilinear wave equations

TL;DR

This paper provides an error analysis for one-stage explicit extended RKN integrators applied to semilinear wave equations, demonstrating optimal second-order convergence without high regularity assumptions.

Contribution

It offers the first error analysis for these integrators on semilinear wave equations, applicable beyond spectral spatial discretizations.

Findings

Proves optimal second-order convergence.

Applies to general spatial discretizations.

Does not require high regularity of solutions.

Abstract

In this paper, we present an error analysis of one-stage explicit extended Runge--Kutta--Nystr\"{o}m integrators for semilinear wave equations. These equations are analysed by using spatial semidiscretizations with periodic boundary conditions in one space dimension. Optimal second-order convergence is proved without requiring Lipschitz continuous and higher regularity of the exact solution. Moreover, the error analysis is not restricted to the spectral semidiscretization in space.