Numerical experiments on the rational Runge-Kutta method

AbstractThe rational Runge-Kutta (RRK) method is a non-linear explicit A-stable scheme for the numerical treatment of initial-value problems for systems of ordinary differential equations (ODEs). The method may be expected to be suitable for the treatment of stiff systems when the number of ODEs is very large. We have studied theoretically and implemented numerically the RRK scheme on problems of the type dx(t)/dt = Ax(t) + w(t), under stiff conditions. We have paid particular attention to the case when the system of ODEs originates from the semi-discretization of an evolution problem for a partial differential equation. The scheme is shown to perform poorly when the eigenvalues of A are widely spread; on the other hand, its performance is reasonable when the eigenvalues of A are “packed” around an average negative value and the forcing term w(t) evolves slowly with respect to this value