Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary PDEs containing such operators and integrated in time with exponential integrators, it is of paramount importance to efficiently approximate actions of $\varphi$-functions of this kind of matrices. In this work, we show how to produce directional split approximations of third order with respect to the time step size. They conveniently employ tensor-matrix products (realized with highly performance level 3 BLAS) and that allow for the effective usage in practice of exponential integrators up to order three. The approach has been successfully tested against state-of-the-art techniques on two well-known physical models, namely FitzHugh--Nagumo and Schnakenberg.

翻译：通过张量积公式对常见多维算子（例如扩散-对流）进行适当离散化，可得到具有$d$维Kronecker和结构的矩阵。对于包含此类算子的演化型偏微分方程，当使用指数积分器进行时间积分时，高效近似此类矩阵的$\varphi$-函数作用至关重要。本文展示了如何构造关于时间步长具有三阶精度的方向分裂近似方法。该方法巧妙利用张量-矩阵乘积（通过高性能三级BLAS实现），使得三阶以内的指数积分器能够有效应用于实际计算。我们通过两个经典物理模型（FitzHugh-Nagumo模型和Schnakenberg模型）将该方法与现有先进技术进行了成功对比测试。