Parallel linear system solvers for Runge-Kutta-Nyström methods

Solving the nonlinear systems arising in implicit Runge-Kutta-Nyström type methods by modified Newton iteration leads to linear systems whose matrix of coefficients is of the form $I - A otimes h^2 J$ where A is the Runge-Kutta-Nyström matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear Newton systems by an inner iteration process. Each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems are of the form $I - B otimes h^2 J$ where B is a nondefective matrix with positive eigenvalues, so that by a similarity transformation, we can decouple the system into subsystems the dimension of which equals the dimension of the system of differential equations. Since the subsystems can be solved in parallel, the resulting integration method is highly efficient on parallel computer systems. The performance of the Parallel Iterative Linear System method for Runge-Kutta-Nyström equations PILSRKN method is illustrated by means of a few examples from the literature.