Exponential integrators for parabolic problems with non-homogeneous boundary conditions

TL;DR

This paper extends exponential integrator methods to handle non-homogeneous boundary conditions in parabolic problems, introducing a correction strategy that preserves convergence order and improves numerical accuracy.

Contribution

It develops a correction technique for exponential Runge-Kutta methods to effectively manage non-homogeneous boundary conditions, maintaining high-order convergence.

Findings

Corrected schemes recover expected convergence orders.

Higher orders up to 2s are achievable with suitable quadrature.

Numerical experiments confirm theoretical convergence results.

Abstract

Exponential Runge-Kutta methods are a well-established tool for the numerical integration of parabolic evolution equations. However, these schemes are typically developed under the assumption of homogeneous boundary conditions. In this paper, we extend classical convergence results to the case of non-homogeneous boundary conditions. Since non-homogeneous boundary conditions typically cause order reduction, we introduce a correction strategy based on smooth extensions of the boundary data. This results in a reformulation as a homogeneous problem with modified source term, to which standard exponential integrators can be applied. For linear problems, we prove that the corrected schemes recover the expected convergence order, and hat higher orders can be attained with suitable quadrature rules, reaching order $2s$ for s-stage Gauss collocation methods. For semilinear problems, our approach preserves the convergence orders guaranteed by exponential Runge-Kutta methods satisfying the corresponding stiff order conditions. Numerical experiments validate the theoretical findings.