IMEX Schemes
Implicit-explicit schemes treat stiff terms implicitly and non-stiff terms explicitly, combining stability with low cost per step.
Splitting by stiffness
Many equations contain both a stiff part (fast diffusion, stiff reactions, fast waves) and a non-stiff part (advection, mild source terms). Treating everything implicitly is stable but expensive because even the easy terms enter the nonlinear solve; treating everything explicitly forces tiny steps because of the stiff part. Implicit-explicit (IMEX) schemes split the difference: advance the stiff terms implicitly for stability and the non-stiff terms explicitly for cheapness.
The result is a step whose stability is set by the non-stiff terms alone, while the stiff terms are handled without the crippling explicit time-step limit, and without the cost of implicitly solving the whole right-hand side.
Construction
IMEX methods pair an implicit scheme (often a diagonally implicit Runge-Kutta or a backward differentiation formula) for the stiff operator with a matched explicit scheme for the non-stiff operator, sharing stage times so the combination retains a stated order of accuracy. Additive Runge-Kutta families are designed specifically for this pairing.
Design considerations
The split must be chosen so the implicit part captures all genuinely stiff dynamics; misclassifying a stiff term as explicit reintroduces a severe step limit. The implicit solve should involve a linear or easily preconditioned operator, which is why diffusion (a linear elliptic operator) is a favorite candidate for implicit treatment.
- Stiff terms implicit, non-stiff terms explicit
- Time step set by the non-stiff dynamics
- Additive Runge-Kutta and BDF-based pairings
- Split must capture all stiffness on the implicit side
IMEX integration is common in reacting-flow, edge-plasma, and MHD models where stiff diffusion or fast parallel dynamics coexist with slower advective transport.