Direct Parallel-in-Time method based on Spectral Deferred Correction

Recent advances in supercomputing architectures have pushed the scientific community to develop more efficient algorithms for high performance computing (HPC).
In particular, the switch from improving processor speed to increasing concurrency in HPC have motivated the development of new parallelization algorithms that can harness the computing power of massively parallel HPC architectures. For two decades, there have been research efforts to develop parallel computing capabilities for time integration methods used for the simulation of time dependent problems like Numerical Weather Prediction, Computational Fluid Dynamics, Fluid-Structure interactions, .... However, to develop such "Parallel-in-Time" algorithms (or PinT), one needs to deal with the sequential nature of time: it is necessary to know the past and the present before computing the future. Hence, computing past, present and future in parallel needs some fundamental changes in the classical computation paradigms used for time-dependent problems.

During this talk, I will present one approach currently developed to perform PinT time-integration for numerical weather prediction. It is based on Spectral Deferred Correction (SDC), a time-integration method that iteratively computes the stages of a fully implicit collocation method using a preconditioned iteration. SDC allows to generate a variety of methods with arbitrary order of accuracy . While there are several parameters that can be used to optimize a SDC method, the main one is the choice of preconditionner. In particular, one can build diagonal SDC preconditioners, either to improve convergence speed or numerical stability for larger time-step size.
One important aspect of those diagonal preconditioners is that they allow computations for SDC iterations to be performed in parallel.

I will present some recent results on building optimized diagonal preconditionner for a split implicit-explicit formulation of SDC and show their application to test problems based on the shallow water equations.