A note on stability of the Douglas splitting method

In this note some stability results are derived for the Douglas splitting method. The relevance of the theoretical results is tested for an advection-reaction equation. 1. Presentation of the results Consider the initial value problem for a system of ODEs u′(t) = F (t, u(t)) (1.1) with 0 ≤ t ≤ T and given initial value u(0). We shall consider numerical schemes with step size τ yielding approximations un to the exact solution u(tn) at time levels tn = nτ for n = 0, 1, 2, · · · , starting with u0 = u(0). For problems that arise by spatial discretization of multi-dimensional PDEs it is often possible to decompose the function F into a number of simpler component functions, F (t, w) = F1(t, w) + F2(t, w) + · · ·+ Fs(t, w). (1.2) Splitting methods use this decomposition by treating in each stage at most one of the components implicitly. The best known method of this type is the ADIPeaceman-Rachford method, but this method can only deal with 2-component splittings, see [5]. In this paper we shall consider the related second-order method of Douglas [1], also known as the method of Stabilizing Corrections [4], v0 = un + τF (tn, un), vi = vi−1 + 12τ ( Fi(tn+1, vi)− Fi(tn, un) ) (i = 1, 2, · · · , s), un+1 = vs, (1.3) with internal vectors vi. A big advantage of (1.3) over many other splitting methods [4, 5] is that all internal vectors vi are consistent approximations to the exact solution, namely at time tn+1. This implies that if we are in a steady state F (u) = 0, with F independent of t, then this steady state is also a stationary point of the scheme (1.3). Received by the editor July 29, 1996. 1991 Mathematics Subject Classification. Primary 65M06, 65M12, 65M20.