Exploiting the Kronecker product structure of φ−functions in exponential integrators. (1st March 2022)

• Record Type: Journal Article
• Authors: Muñoz‐Matute, Judit; Pardo, David; Calo, Victor M.
• Is Part Of: International journal for numerical methods in engineering. Volume 123:Number 9(2022)
• Page Start: 2142
• Page End: 2161
• Publication Date: 2022-03-01
• Subjects: φ−functions -- exponential integrators -- finite element method -- Kronecker sum -- semilinear parabolic problems -- Sylvester equation; Numerical analysis -- Periodicals; Engineering mathematics -- Periodicals; 620.001518
• DOI: 10.1002/nme.6929
• Languages: English
• ISSNs: 0029-5981

Abstract

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.