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 hardware and software architectures.

翻译：通过张量积公式对常见多维算子（例如扩散或扩散-对流算子）进行适当离散化，会得到具有$d$维Kronecker和结构的矩阵。对于包含此类算子并采用指数积分器进行时间积分的演化偏微分方程，高效逼近所生成矩阵的$\varphi$函数作用量至关重要。本工作展示了如何构建关于时间步长具有三阶精度的定向分裂逼近方法。该方法巧妙地运用张量-矩阵乘积（即所谓的$\mu$模态乘积及相关Tucker算子，实际计算中采用高性能三级BLAS实现），使得指数Runge--Kutta积分器可有效应用于高达三阶精度的计算。该技术亦能在现代计算硬件（如图形处理器）上高效实现。本方法已通过二维Schnakenberg系统与三维FitzHugh--Nagumo系统这两个能产生图灵模式的经典物理模型，在不同软硬件架构上与前沿技术进行了成功对比验证。