Efficient numerical integration for N th-order non-autonomous linear differential equations

Abstract

We consider the numerical integration of high-order linear non-homogeneous differential equations, written as first order homogeneous linear equations, and using exponential methods. Integrators like Magnus expansions or commutator-free methods belong to the class of exponential methods showing high accuracy on stiff or oscillatory problems, but the computation of the exponentials or their action on vectors can be computationally costly. The first order differential equations to be solved present a special algebraic structure (associated with the companion matrix) which allows to build new methods (hybrid methods between Magnus and commutator-free methods). The new methods are of similar accuracy as standard exponential methods with a reduced complexity. Additional parameters can be included into the scheme for optimization purposes. We illustrate how these methods can be obtained and present several sixth-order methods which are tested in several numerical experiments.