2023/10/11 by Fabio Cassini, Cassini, Fabio
Engineering · Mathematics · #FOS: Mathematics #Frequency Control in Power Systems #Numerical Analysis (math.NA) #Numerical methods for differential equations #Power System Optimization and Stability
paper · pdf · doi:10.48550/arxiv.2310.07551
openalex publication_date 2023/10/11 · openalex created_date 2023/10/13 · openalex updated_date 2026/07/28
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 φ-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 μ-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.