Contractivity of waveform relaxation Runge-Kutta iterations and related limit methods for dissipative systems in the maximum norm

Contractivity properties of Runge-Kutta methods are analyzed, with suitable interpolation implemented using waveform relaxation strategy for systems of ordinary differential equations that are dissipative in the maximum norm. In general, this type of implementation, which is quite appropriate in a parallel computing environment, improves the stability properties of Runge-Kutta methods. As a result of this analysis, a new class of methods is determined, which is different from Runge-Kutta methods but closely related to them, and which combines its high order of accuracy and unconditional contractivity in the maximum norm. This is not possible for classical Runge-Kutta methods.