Runge-Kutta Methods
Runge-Kutta methods build high-order time integration from several intermediate evaluations of the right-hand side within a single step.
Multiple stages, one step
Runge-Kutta (RK) methods advance an ordinary differential equation over one time step by sampling the right-hand side at several intermediate stages and combining them with carefully chosen weights. The classical fourth-order method (RK4) uses four evaluations and cancels error terms up to fourth order in the step size, giving high accuracy from a self-contained single step that needs no history.
This one-step nature makes RK methods easy to start, easy to change step size, and well suited to adaptive control, in contrast to multistep methods that depend on several previous values.
Butcher tableaux and order
An RK method is specified compactly by a Butcher tableau of stage coefficients and weights. Explicit RK methods have a strictly lower-triangular coefficient matrix so each stage depends only on earlier ones; diagonally implicit (DIRK) and fully implicit RK methods include the current stage, requiring solves but gaining stability for stiff problems.
Embedded pairs and stability
Embedded RK methods compute two solutions of different orders from the same stages; their difference estimates the local error and drives adaptive step-size control. For hyperbolic conservation laws, strong-stability-preserving (SSP) RK methods keep the nonlinear stability properties of the underlying spatial scheme, preventing spurious oscillations.
- Butcher tableau encodes stages, weights, and nodes
- Explicit RK for non-stiff, implicit/DIRK for stiff problems
- Embedded pairs give cheap local error estimates
- SSP variants preserve nonlinear stability for shocks
Runge-Kutta integrators are ubiquitous across fusion codes, from orbit following and turbulence to MHD, chosen in explicit, implicit, or SSP form to match the stiffness and stability demands of the problem.