In this paper we consider an approach to improve the performance of exponential integrators/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 advection-diffusion-reaction equations for which we are then able to compute the required matrix functions efficiently. Both a linear stability analysis and numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also propose new Lawson type integrators that further improve on these stability properties. The effectiveness of the approach is highlighted by a number of numerical examples in two and three space dimensions.
翻译:在本文中,我们考虑一种方法,以改善指数积分器/劳森方案的性能,在相关但通常更简单的问题的解可有效计算时。虽然对于隐式方法而言这种方法很常见(例如通过使用预处理器),但对于指数积分器来说,这被证明更具挑战性。在此,我们建议从对流-扩散-反应方程中提取一个常数系数微分算子,然后我们能够有效地计算所需的矩阵函数。线性稳定性分析和数值实验表明,得到的方案可以无条件稳定。事实上,我们发现指数积分器和劳森方案可以具有比类似构造的隐式-显式方案更好的稳定性质。我们还提出了新的劳森类型积分器来进一步改善这些稳定性质。该方法的有效性通过二维和三维空间中的许多数值示例突出体现。