Spectral Deferred Correction
Spectral deferred correction builds high-order time integrators by iteratively correcting a low-order solution against a spectral quadrature of the error.
Iterating toward high order
Spectral deferred correction (SDC) is a systematic way to construct arbitrarily high-order time integrators from simple low-order ones. Within each time step it places several intermediate substep nodes, usually Gauss or Radau points, and first computes a provisional solution with a cheap low-order method such as backward or forward Euler. It then repeatedly solves a correction equation for the error, using a high-order spectral quadrature to integrate the residual accurately. Each correction sweep raises the order of accuracy by (roughly) one, up to the accuracy the node set can support.
This turns a well-understood first-order building block into a high-order method without deriving complicated multi-stage coefficients, and it does so with a uniform, extensible structure.
Why the quadrature matters
The correction is driven by the integral of the residual over the step. Evaluating that integral on high-order spectral quadrature nodes is what gives the method its accuracy; a low-order quadrature would cap the achievable order. The iteration converges to the collocation solution defined by the node set, which is a high-order implicit Runge-Kutta method.
Variants and appeal
SDC extends naturally to stiff problems (using implicit sweeps), to IMEX splittings for mixed stiff and non-stiff terms, and to parallel-in-time integration through the PFASST algorithm, which couples SDC sweeps across time slices with a multigrid hierarchy. Its flexibility, easy adjustment of order, and clean handling of stiffness and splitting make it attractive for multi-physics time stepping.
- Low-order predictor corrected by spectral-quadrature sweeps
- Each sweep raises the order by about one
- Converges to a high-order collocation (implicit RK) method
- Extends to stiff, IMEX, and parallel-in-time settings
SDC and its parallel variants offer a route to high-order, stable time integration for stiff multi-physics fusion models where deriving bespoke high-order schemes would be cumbersome.