On the convergence of Lawson methods for semilinear stiff problems
This work clarifies a long-standing discrepancy in the convergence theory of Lawson methods for stiff evolution equations, benefiting numerical analysts and practitioners using exponential integrators.
The paper explains why Lawson methods, despite lacking stiff order conditions, can achieve high-order convergence for semilinear stiff problems under appropriate regularity assumptions, such as periodic boundary conditions, but may suffer order reduction for Dirichlet boundary conditions. Numerical examples for the Schrödinger equation are provided.
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 Schrödinger equation are included.