Adaptive Time Domain Decomposition for systems of ODEs on Grid architecture

摘要

Modelling complex systems can lead to solve ODEs, and/or DAEs systems to understand the dynamics behavior, eventually stiff, of the solutions. The main difficulty in term of parallel implementation of such systems of equations is the poor granularity of the computations, especially, for the grid computing architecture, which is characterized by a large number of processors. Nevertheless, the development of low cost parallel computer change the concept of an efficient parallel implementation. The number of available processors allows to consider these computational resources to improve the confidence in the solution by combining several schemes of different orders and different discretisations, to validate and to verify the result. The main approach used in the past to obtain parallel solver for ODEs systems consists to parallelize "across the method" which distributes to the processors the computation of steps of multi-step methods as Runge-Kutta RK(4) method. The number of processors used is limited as it depends of the number of steps. An another approach is the time decomposition method was originaly introduced by people from the multi-grid field and was applied to solve PDEs. The "parareal" scheme proposed in [5] follows this approach to consider two levels of grids in time in order to split the domain in time in subdomains. A prediction of the solution is computed on the fine grid in time in parallel. Then at each end boundaries of time sub-domains the solution makes a jump with the previous initial boundary value (IBV) of the next subdomain in time. A correction of the IBV for the next fine grid iterate is then computed on the coarse grid in time. [2] shows that the method converge at least in a finite number of iterations, due to the propagation of the fine time grid solution as at each iteration k, the IBV of the k~(th) time sub-domain is exact. Nevertheless, this approach seems to have difficulties for stiff non linear problems eventually with bifurcation parameters. We propose in this paper to investigate the same concept of prediction on fine grid and correction on coarse grid but with introducing more adaptivity in the definition of the fineness of the grids, the number of grids for correction, and the decomposition of the time interval. We expect to improve the method in order to solve stiff ODEs problems. We postulate that we can have as much processors as we need (grid architecture), and the main focus is to reduce the elapse time to obtain the solution on a finite time interval with a given precision.