A new class of second order linearly implicit fractional step methods

Abstract

A new family of linearly implicit fractional step methods is proposed and analysed in this paper. The combination of one of these time integrators with a suitable spatial discretization permits a very efficient numerical solution of semilinear parabolic problems. The main quality of this new family of methods, compared to other existing time integrators of this type, is that they are stable when the spatial differential operator is decomposed in a number m of “simpler” operators which do not necessarily commute. We prove that these methods satisfy this general stability result as well as they are second order consistent. Both consistency and stability are proven for an operator splitting in an arbitrary number m of terms ( m ⩾ 2 ). Finally, a numerical experiment illustrates these theoretical results in the last section of the paper.