A generalization of Peaceman–Rachford fractional step method

In this paper we develop a set of time integrators of type fractional step Runge–Kutta methods which generalize the time integrator involved in the classical Peaceman–Rachford scheme. Combining a time semidiscretization of this type with a standard spatial discretization, we obtain a totally discrete algorithm capable of discretizing efficiently a general parabolic problem if suitable splittings of the elliptic operator are considered. We prove that our proposal is second order consistent and stable even for an operator splitting in m terms which do not necessarily commute. Finally, we illustrate the theoretical results with various applications such as alternating directions or evolutionary domain decomposition.