Operator Splitting Methods

Suppose one has an evolution equation of a general form u — A1 + A2 [22]

where each of A1 and A2 may depend on x, t, u, 8xu, etc. The criterion by which the RHS of [22] has been broken down into the two terms is this: each of the auxiliary problems u = Ak, k = 1, 2

can be solved easily. Then, to approximate the solution of [22], one solves the two individual equations [23] in sequence over each time step. The global error generated by this splitting is 0(t), in addition to any errors that may be introduced when solving each of the [23]. More accurate versions of this method are also available. However, we do not advocate for operator-splitting methods because of their stability properties. Namely, although for linear equations such methods are unconditionally stable, for nonlinear equations, in general, the stability of operator-splitting methods is determined by a condition similar to [8], even if each of eqns [23] is solved exactly. Hence, in such cases, operator-splitting methods offer no advantage over explicit methods. Operator-splitting methods find their primary use in solving (nonlinear) wave equations without dissipation, which, however, do not normally occur in ecological models.

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