Operator Splitting
Operator splitting advances a multi-physics problem by solving its constituent operators in sequence, each with a method suited to it.
Divide the physics, not the domain
When an evolution equation combines several physical processes, such as advection, diffusion, and reaction, operator splitting advances each process in turn over a time step rather than solving them all together. Each sub-operator can then use the integrator best matched to it: an explicit upwind scheme for advection, an implicit solve for stiff diffusion, a specialized ODE solver for reactions.
This modularity is the method's main appeal: complex coupled systems become sequences of well-understood, independently optimized sub-solves, and existing single-physics solvers can be reused.
Lie and Strang splitting
The simplest scheme, Lie (or Godunov) splitting, applies the operators sequentially and is first-order accurate in time; the splitting error comes from the fact that the operators do not commute. Strang splitting recovers second order by using a symmetric sequence: half a step of the first operator, a full step of the second, then another half step of the first.
Splitting error and pitfalls
The order reduction from splitting is real and can matter near boundaries or steady states, where the split solution may not satisfy the coupled balance the full system would. Careful choice of ordering, symmetric composition, and consistent boundary treatment across sub-steps limit these errors. For stiff coupled terms, splitting can introduce spurious transients that require attention.
- Each sub-operator solved with its best-suited method
- Lie splitting first order; Strang splitting second order
- Splitting error from non-commuting operators
- Boundary and steady-state consistency need care
Operator splitting is pervasive in multi-physics fusion modeling, letting transport, atomic-physics, and field-evolution modules be developed and validated separately, then composed into a coupled simulation.