A ⋆‐product solver with spectral accuracy for non‐autonomous ordinary differential equations

Abstract

AbstractA new method for solving non‐autonomous ordinary differential equations is proposed, the method achieves spectral accuracy. It is based on a new result which expresses the solution of such ODEs as an element in the so called ⋆‐algebra. This algebra is equipped with a product, the ⋆‐product, which is the integral over the usual product of two bivariate distributions. Expanding the bivariate distributions in bases of Legendre polynomials leads to a discretization of the ⋆‐product and this allows for the solution to be approximated by a vector that is obtained by solving a linear system of equations. The effectiveness of this approach is illustrated with numerical experiments.