Operator Splitting
Advancing a multi-physics problem by solving each physical process in sequence over each time step, using the best method for each.
Divide the physics, not the domain
Many evolution equations combine several distinct processes, for instance advection plus diffusion plus reaction. Operator splitting advances each process separately in turn over a time step, applying whatever specialized and efficient integrator suits each one, rather than building a single monolithic scheme for the whole coupled operator. This modularity lets a code reuse a fast explicit advection solver alongside an implicit diffusion solver.
Lie and Strang splitting
The simplest scheme, Lie (first-order) splitting, applies each operator for a full step in sequence. It is only first-order accurate because the operators generally do not commute, and solving them sequentially neglects their interaction. Strang splitting recovers second order by using a symmetric sequence: apply the first operator for half a step, the second for a full step, then the first for another half step. The symmetry cancels the leading splitting error.
- Lie splitting: sequential full steps, first-order accurate
- Strang splitting: symmetric half-full-half sequence, second-order accurate
- Each sub-step can use a method tailored to that operator's stiffness and structure
The splitting error
The price of splitting is the splitting error, which arises because the true solution mixes the processes continuously while the split scheme treats them one at a time. This error is proportional to the commutator of the operators: if the processes nearly commute, splitting is very accurate; if they interact strongly, the error can be significant and the time step must shrink. Splitting can also disturb steady states, so care is needed near equilibrium.
Use in practice
Operator splitting is ubiquitous in atmospheric chemistry, combustion, and plasma transport, where advection, diffusion, and reaction have very different numerical character. It is closely related to IMEX schemes, which also isolate stiff from non-stiff physics, but splitting solves each part fully and separately rather than combining them in one tableau. The choice between them depends on how strongly the processes are coupled.