Implicit-Explicit (IMEX) Schemes
Time integrators that treat stiff terms implicitly and non-stiff terms explicitly, avoiding both instability and costly nonlinear solves.
Splitting by stiffness
Many evolution equations combine a stiff term (fast diffusion, fast waves, or a rapidly relaxing reaction) with a non-stiff term (slower advection or a mild nonlinearity). A fully explicit method would need a tiny time step to keep the stiff term stable. A fully implicit method would be stable but require solving a large nonlinear system every step. IMEX schemes split the difference: the stiff part is integrated implicitly for stability, the non-stiff part explicitly for cheapness.
The canonical example
For an equation du/dt = F(u) + G(u), where F is non-stiff and G is stiff, an IMEX scheme evaluates F at known past values but G at the new time level. Often the stiff G is linear (a diffusion operator), so the implicit step reduces to a single linear solve per step rather than a nonlinear iteration. This is the pattern behind implicit-diffusion, explicit-advection solvers used in transport codes.
Building blocks
- IMEX multistep methods pair an implicit backward-differentiation formula with an explicit Adams or extrapolation formula
- IMEX Runge-Kutta methods use a paired tableau, one explicit and one diagonally-implicit, sharing stage times
- Order conditions must be satisfied jointly by the implicit and explicit parts to reach the target order
Design trade-offs
The stability of an IMEX method depends on the interaction between the two parts, so the split must genuinely isolate the stiffness; a poorly chosen split can be less stable than either pure method. IMEX schemes are widely used in plasma transport, reacting flows, and climate models, wherever fast and slow physics coexist and a global implicit solve over everything would be too expensive. They complement exponential integrators, which take a different route to the same stiff-term problem.