Exponential integrators for coupled self-adjoint non-autonomous partial differential systems

We consider the numerical integration of coupled self-adjoint non-autonomous partial differential systems. Under convergence conditions, the solution can be written as a series expansion where each of its terms correspond to solutions of linear time dependent matrix differential equations with oscillatory solutions that must be solved numerically. In this work, we analyze second order of Magnus integrators whose numerical error grows with the number of terms considered in the truncated series, n, at a rate that still allows us to guarantee convergence of the numerical series. In addition, the integrator can be implemented with a recursive algorithm such that the computational cost of the method grows only linearly with the number of terms of the series. Higher order Magnus integrators are also analyzed. Commutator-free Magnus integrators can be used with a similar recursive algorithm and can provide highly accurate results, but they show a faster error growth with n, and some caution must be taken if these methods are used. Numerical experiments confirm the performance of the proposed algorithm.