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For many PDE problems in higher space dimension, such as the advectiondiffusion-reaction system Ut

+ 'V. (gu) =

'V. (D'Vu)

+ f(u) ,

it is in general inefficient or infeasible to apply one and the same integration formula to the different parts of the system. For example, the chemistry can be very stiff, which calls for an implicit ODE method. On the other hand, if the advection is discretized in space using a limiter, then explicit methods are often much more suitable for that part of the equation. Moreover, use of a single implicit integration formula for the whole problem readily leads to a nonlinear algebraic system too large to handle due to the simultaneous coupling over the species and over space. In such cases a more tuned approach based on an appropriate form of splitting is advocated. The general idea behind splitting is breaking down a complicated problem into smaller parts for the sake of time stepping, such that the different parts can be solved efficiently with suitable integration formulas.

1 Operator Splitting This section is devoted to the technique called operator splitting or time splitting. We will discuss this form of splitting for ODE and PDE problems without invoking actual integration formulas. Hence in this section we mainly focus on concepts rather than on actual methods.

1.1 First-Order Splitting Linear ODE Problems Let us first illustrate the notion of splitting by considering a linear, homogeneous ODE system

w'(t) = Aw(t) , t > 0,

w(O) = wo,

(1.1)

W. Hundsdorfer et al., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations © Springer-Verlag Berlin Heidelberg 2003

326

IV Splitting Methods

and assume for A a two-term splitting

System (1.1) may for example be seen as a semi-discretization of a linear PDE problem with homogeneous or periodic boundary conditions. The solution of (1.1) is given by (1.2) where T = t n+1 - tn. If we wish to use only Al and A2 separately, instead of the full A, then (1.2) can be approximated by (1.3) with Wn approximating w(tn). This is the simplest splitting, in which we solve the two subproblems

ftw*(t) = AIW*(t)

for tn < t ~ tn+1 with w*(tn ) = wn ,

one after another, starting from wn , and take Wn+1 = w**(tn+1) to complete the splitting integration step. Replacing (1.2) by (1.3) normally introduces an error, the so-called splitting error. Inserting the exact solution w of the original problem into (1.3) gives w(tn+1) = e TA2 e TA1 w(tn) + TPn , with local truncation error Pn. Recall that TPn is the error introduced per step starting from the true solution, hence it is the local splitting error. We have eTA = 1+ T(AI + A 2) + ~T2(AI + A2)2 + ... , eTA2eTAl

= 1+ T(AI + A 2) + ~T2(A~ + 2A2AI + A~) + ....

The truncation error thus satisfies

Pn

(1.4)

with (1.5) being the commutator of Al and A 2. Obviously (1.3) is a first-order process unless Al and A2 commute. In case Al and A2 do commute we have

which can be seen by using the power series expansion (1.2.19) for the exponential function. It follows that for commuting