We present a method for computing actions of the exponential-like $\varphi$-functions for a Kronecker sum $K$ of $d$ arbitrary matrices $A_\mu$. It is based on the approximation of the integral representation of the $\varphi$-functions by Gaussian quadrature formulas combined with a scaling and squaring technique. The resulting algorithm, which we call PHIKS, evaluates the required actions by means of $\mu$-mode products involving exponentials of the small sized matrices $A_\mu$, without forming the large sized matrix $K$ itself. PHIKS, which profits from the highly efficient level 3 BLAS, is designed to compute different $\varphi$-functions applied on the same vector or a linear combination of actions of $\varphi$-functions applied on different vectors. In addition, thanks to the underlying scaling and squaring techniques, the desired quantities are available simultaneously at suitable time scales. All these features allow the effective usage of PHIKS in the exponential integration context. In fact, our newly designed method has been tested on popular exponential Runge--Kutta integrators of stiff order from one to four, in comparison with state-of-the-art algorithms for computing actions of $\varphi$-functions. The numerical experiments with discretized semilinear evolutionary 2D or 3D advection--diffusion--reaction, Allen--Cahn, and Brusselator equations show the superiority of the proposed $\mu$-mode approach.

翻译：摘要：针对任意矩阵$A_\mu$的Kronecker和$K$，我们提出了一种计算类指数$\varphi$-函数作用的方法。该方法基于高斯求积公式对$\varphi$-函数积分表示的逼近，并结合了缩放与平方技术。由此产生的算法（称为PHIKS）通过涉及小型矩阵$A_\mu$指数的$\mu$-模式乘积来计算所需作用，无需直接构建大型矩阵$K$本身。PHIKS利用高效的BLAS三级库，专为计算同一向量上的不同$\varphi$-函数作用，或不同向量上$\varphi$-函数作用的线性组合而设计。此外，得益于底层缩放与平方技术，所需量可在合适的时间尺度上同时获得。这些特性使PHIKS在指数积分框架中具有高效适用性。我们已将该新方法应用于刚度阶数为一至四的经典指数龙格-库塔积分器，并与当前最先进的$\varphi$-函数作用计算算法进行对比。基于离散化半线性演化二维/三维对流-扩散-反应方程、Allen-Cahn方程及Brusselator方程的数值实验表明，所提出的$\mu$-模式方法具有优越性。