High-order time stepping methods based on spectral deferred corrections

Daniel Ruprecht (The University of Leeds)


Spectral deferred corrections (SDC) are an iterative approach to constructing numerical time stepping methods of high order. They can be considered as a framework for generating high order schemes out of a low order base method by iteratively solving for the stages of collocation methods. Their advantage stems from the fact that problem-tailored features of the base method (splitting, multi-rate, etc), which are often difficult to achieve for high order methods, can be retained.

I will present two new integration methods based on SDC. The first generalises the second order Verlet-type Boris integrator, which is the de-facto standard for computing trajectories of charged particles in electro-magnetic fields. The resulting Boris-SDC allows to generalise the intrinsically second order Boris method to arbitrary high order. The second is SDC with fast-wave slow-wave splitting (fwsw-sdc). In compressible flow, the existence of fast acoustic waves often leads to numerical difficulties that can be overcome by implicit-explicit splitting. FWSW-SDC uses an implicit-explicit Euler as based method, treating stiff, fast acoustic waves implicitly and slower modes explicitly. Theoretical analysis shows that is has favourable dispersion properties while numerical examples show its competitiveness with IMEX Runge-Kutta methods.

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