Stability of Runge-Kutta methods used in modular integration

A pair of Runge-Kutta methods is applied to a system of ordinary differential equations in a modular fashion known as time point relaxation. For a class of two by two linear systems with constant coefficients, the concept of coupling stability is introduced. This is one way of measuring the loss of stability due to the decoupling of the system into two scalar subsystems. The strategy for handling the interactions between the two modules is controlled by a parameter, where certain choices of the parameter correspond to the Gauss-Jacobi and Gauss-Seidel method. Results are obtained for the case when Runge-Kutta methods in general are applied with only one iteration per time-step. The case with several iterations is investigated for the well-known θ-methods.