On the convergence of Lawson methods for semilinear stiff problems

TL;DR

This paper investigates the convergence properties of Lawson methods for stiff evolution equations, revealing that under certain regularity conditions, they achieve high-order convergence despite lacking stiff order conditions.

Contribution

It provides a detailed analysis of the regularity assumptions needed for high-order convergence of Lawson methods, explaining their effectiveness in certain boundary condition scenarios.

Findings

High-order convergence is achievable under specific regularity assumptions.

Lawson methods perform well with periodic boundary conditions.

Order reduction occurs with Dirichlet boundary conditions.

Abstract

Since their introduction in 1967, Lawson methods have achieved constant interest in the time discretization of evolution equations. The methods were originally devised for the numerical solution of stiff differential equations. Meanwhile, they constitute a well-established class of exponential integrators. The popularity of Lawson methods is in some contrast to the fact that they may have a bad convergence behaviour, since they do not satisfy any of the stiff order conditions. The aim of this paper is to explain this discrepancy. It is shown that non-stiff order conditions together with appropriate regularity assumptions imply high-order convergence of Lawson methods. Note, however, that the term regularity here includes the behaviour of the solution at the boundary. For instance, Lawson methods will behave well in the case of periodic boundary conditions, but they will show a dramatic order reduction for, e.g., Dirichlet boundary conditions. The precise regularity assumptions required for high-order convergence are worked out in this paper and related to the corresponding assumptions for splitting schemes. In contrast to previous work, the analysis is based on expansions of the exact and the numerical solution along the flow of the homogeneous problem. Numerical examples for the Schr\"odinger equation are included.