In this paper we consider an approach to improve the performance of exponential Runge--Kutta integrators and Lawson schemes} in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for implicit methods such an approach is common (e.g. by using preconditioners), for exponential integrators this has proven more challenging. Here we propose to extract a constant coefficient differential operator from the semilinear advection-diffusion-reaction equation for which, in many situations, efficient methods are known to compute the required matrix functions. Both a linear stability analysis and {\color{black} extensive} numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators of Runge--Kutta type and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also derive two new Lawson type integrators that further improve on these stability properties. The overall effectiveness of the approach is highlighted by a number of performance comparisons on examples in two and three space dimensions.

翻译：本文考虑一种方法，在可高效计算相关但通常更简单问题的解时，提升指数龙格-库塔积分器和Lawson格式的性能。虽然对于隐式方法而言，此类方法（如使用预处理技术）较为常见，但对于指数积分器则更具挑战性。我们提出从半线性对流-扩散-反应方程中提取常系数微分算子，在许多情况下，已有高效方法可计算所需的矩阵函数。线性稳定性分析与大量数值实验均表明，所得格式可保持无条件稳定性。事实上，我们发现龙格-库塔型指数积分器和Lawson格式的稳定性优于类似构造的隐式-显式格式。此外，我们推导出两种新型Lawson积分器，进一步改善了这些稳定性特性。通过二维与三维空间算例的性能对比，充分证明了该方法的整体有效性。