Explicit exponential Runge-Kutta methods for semilinear integro-differential equations

TL;DR

This paper develops explicit exponential Runge-Kutta methods for semilinear integro-differential equations, providing order conditions, convergence analysis, and numerical validation in a Hilbert space framework.

Contribution

It introduces new explicit exponential Runge-Kutta methods with proven convergence for semilinear integro-differential equations, including order conditions and spectral Galerkin discretization.

Findings

Derived order conditions for general order p

Constructed methods satisfying order 1 and 2 conditions

Numerical experiments confirm theoretical convergence results

Abstract

The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order $p$ and prove convergence of order $p$, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.