In this manuscript, we propose an efficient, practical and easy-to-implement way to approximate actions of $\varphi$-functions for matrices with $d$-dimensional Kronecker sum structure in the context of exponential integrators up to second order. The method is based on a direction splitting of the involved matrix functions, which lets us exploit the highly efficient level 3 BLAS for the actual computation of the required actions in a $\mu$-mode fashion. The approach has been successfully tested on two- and three-dimensional problems with various exponential integrators, resulting in a consistent speedup with respect to a technique designed to compute actions of $\varphi$-functions for Kronecker sums.

翻译：本文提出了一种高效、实用且易于实现的方法，用于近似计算包含$d$维Kronecker和结构的矩阵的$\varphi$-函数作用，适用于最高二阶的指数积分器。该方法基于所涉及矩阵函数的方向分裂，使得我们能够以$\mu$-模方式利用高效的三级BLAS库进行所需作用的实际计算。该方案已在二维和三维问题中通过多种指数积分器成功测试，与专为Kronecker和计算$\varphi$-函数作用的设计方法相比，始终获得一致的加速效果。