Exploiting the Kronecker product structure of φ−functions in exponential integrators. (1st March 2022) - Muñoz‐Matute, Judit; Pardo, David; Calo, Victor M.
Abstract: Exponential time integrators are well‐established discretization methods for time semilinear systems of ordinary differential equations. These methods use φ − functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the φ − function action over vectors for two‐dimensional (2D) matrices expressed as a Kronecker sum. For that, we present an auxiliary exponential‐related matrix function that we express using Kronecker products of one‐dimensional matrices. We exploit state‐of‐the‐art implementations of φ − functions to compute this auxiliary function's action and then recover the original φ − action by solving a Sylvester equation system. Our approach allows us to save memory and solve exponential integrators of 2D+time problems in a fraction of the time traditional methods need. We analyze the method's performance considering different linear operators and with the nonlinear 2D+time Allen–Cahn equation.