Exploiting the Kronecker product structure of φ −functions in exponential integrators

Exponential time integrators are well-established discretization methods for time semilinear systems of ordinary differential equations. These methods use (Formula presented.) functions, which are matrix functions related to the exponential. This work introduces an algorithm to speed up the computation of the (Formula presented.) 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 (Formula presented.) functions to compute this auxiliary function's action and then recover the original (Formula presented.) 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.