Time-parallelisation techniques with applications to plasma simulation

Federico Danieli

Abstract: Parallel-in-time (PinT) methods have become an increasingly popular tool in the numerical solution of time-dependent Partial Differential Equations (PDEs), in light of their potential for extracting additional concurrency when spatial parallelism saturates. Particularly complex applications, such as the ones involving plasma simulations for fusion research, would benefit greatly from the implementation of such methods, given the large computational time usually necessary for the numerical solution of their governing equations. In the first part of this thesis, we develop the concept of Space-Time Block Preconditioning (STBP), a technique for simplifying the task of time-parallelising the solution of a whole system of PDEs by reducing it to that of inverting space-time operators acting on single variables only, which are more amenable to be tackled by PinT algorithms. This technique borrows from block preconditioning strategies developed for spatial operators in the framework of time-stepping algorithms, and extends them to the whole space-time setting. After illustrating the guiding principles behind STBP, we measure its effectiveness with applications in fluid dynamics and magnetohydrodynamics. In both cases, the resulting pre-conditioners show excellent scaling properties with respect to mesh refinement, and their applications come at little-to-no overhead cost in terms of iterations to convergence with respect to their time-stepping counterpart, thus providing evidence for their potential parallel efficiency. Ultimately, however, the parallel performance of a space-time block preconditioner depends on that of its internal components, which still rely on already-existing PinT methods. This might become an issue when dealing with advection-dominated or hyperbolic equations, as classical PinT algorithms have shown to struggle in this case. For this reason, the second part of this thesis is dedicated to investigating the causes behind this degradation in performance, as well as to study more recent PinT approaches for overcoming this limitation.