Suitable discretizations through tensor product formulas of popular multidimensional operators (diffusion or diffusion--advection, for instance) lead to matrices with $d$-dimensional Kronecker sum structure. For evolutionary Partial Differential Equations containing such operators and integrated in time with exponential integrators, it is then of paramount importance to efficiently approximate the actions of $\varphi$-functions of the arising 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 (the so-called $\mu$-mode product and related Tucker operator, realized in practice with high performance level 3 BLAS), and allow for the effective usage of exponential Runge--Kutta integrators up to order three. The technique can also be efficiently implemented on modern computer hardware such as Graphic Processing Units. The approach has been successfully tested against state-of-the-art techniques on two well-known physical models that lead to Turing patterns, namely the 2D Schnakenberg and the 3D FitzHugh--Nagumo systems, on different architectures.

翻译：通过张量积公式对常见多维算子（例如扩散或扩散-对流）进行合适的离散化，可得到具有d维Kronecker和结构的矩阵。对于包含此类算子并使用指数积分器进行时间积分的演化偏微分方程，高效逼近由此产生的矩阵的φ-函数作用至关重要。本文展示了如何构造相对于时间步长具有三阶精度的方向分裂逼近。该方法巧妙地利用张量-矩阵乘积（即所谓的μ-模乘积及相关的Tucker算子，实际中通过高性能3级BLAS实现），并支持有效使用高达三阶的指数龙格-库塔积分器。该技术还可高效实现于现代计算机硬件（如图形处理单元）。我们针对两个产生图灵斑图的经典物理模型（即二维Schnakenberg系统和三维FitzHugh-Nagumo系统），在不同架构上与现有先进技术进行了对比测试，验证了该方法的有效性。