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, which has turned out to be competitive for solving space discretizations of certain types of partial differential equations. 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.

中文翻译：

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半线性刚性问题Lawson方法的收敛性

自 1967 年引入以来，Lawson 方法一直对演化方程的时间离散化产生兴趣。这些方法最初是为刚性微分方程的数值解而设计的。同时，它们构成了一类完善的指数积分器，已证明在求解某些类型的偏微分方程的空间离散化方面具有竞争力。Lawson 方法的流行与它们可能具有不良收敛行为的事实形成了某种对比，因为它们不满足任何严格的顺序条件。本文的目的是解释这种差异。结果表明，非刚性阶条件和适当的正则性假设意味着劳森方法的高阶收敛性。但是请注意，这里的术语正则性包括解在边界处的行为。例如，Lawson 方法在周期性边界条件的情况下表现良好，但对于例如 Dirichlet 边界条件，它们将显示出显着的阶数减少。本文提出了高阶收敛所需的精确正则性假设，并与分裂方案的相应假设相关。与以前的工作相比，该分析基于沿齐次问题流的精确解和数值解的扩展。包括薛定谔方程的数值例子。Dirichlet 边界条件。本文提出了高阶收敛所需的精确正则性假设，并与分裂方案的相应假设相关。与以前的工作相比，该分析基于沿齐次问题流的精确解和数值解的扩展。包括薛定谔方程的数值例子。Dirichlet 边界条件。本文提出了高阶收敛所需的精确正则性假设，并与分裂方案的相应假设相关。与以前的工作相比，该分析基于沿齐次问题流的精确解和数值解的扩展。包括薛定谔方程的数值例子。