A Generalization of Peaceman & Rachford Fractional Step Method

It is well known that, in the numerical resolution of linear time dependent coefficient parabolic problems, the use of Fractional Step methods provides nu-merical algorithms which are more efficient than other classical schemes. Such advantage is obtained by using certain splittings of the spatial elliptic operator as a sum of “simpler ” operators joint to certain special time integrators where, in each fractionary step, only one addend of the splitting acts implicitly. Revis-ing classical literature concerning Fractional Step methods (see compendia [1] and [4]) there are only two classical schemes which satisfy suitable absolute sta-bility properties for the case of an operator splitting in an arbitrary number m of addends that do not necessarily commute. They are the Fractionary Implicit Euler scheme (with m implicit stages), which is convergent of first order, and the two-cycle method (with 2m implicit stages), which reaches second order of convergence. In this paper, we develop a second order stable method with 2m − 2 implicit stages which has two main advantages with respect to the two-cycle scheme: it i