Exploiting the Kronecker product structure of <i>?</i> ?functions in exponential integrators

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