Toward an efficient parallel in time method for partial differential equations

Abstract

A new method for the parallelization of numerical methods for partial differential equations (PDEs) in the temporal direction is presented. The method is iterative with each iteration consisting of deferred correction sweeps performed alternately on fine and coarse space-time discretizations. The coarse grid problems are formulated using a space-time analog of the full approximation scheme popular in multigrid methods for nonlinear equations. The current approach is intended to provide an additional avenue for parallelization for PDE simulations that are already saturated in the spatial dimensions. Numerical results and timings on PDEs in one, two, and three space dimensions demonstrate the potential for the approach to provide efficient parallelization in the temporal direction.

Integrating an N-Body Problem with SDC and PFASST

“…When run on one time-processors, PFASST reduces to a multi-level SDC scheme. Comparing SDC (12,7) and PFASST (8,7,1) we note that by switching from a single-level SDC scheme of a given order to a multi-level SDC scheme with comparable accuracy, the number of iterations are reduced (from 12 to 8 in our case). On the other hand, significant additional costs are introduced due to the additional f evaluations required by the coarse step and the transfer operations between fine and coarse quadrature nodes.…”

“…However, the extra computational work of the multi-level SDC scheme can be distributed across multiple processors as demonstrated in the three remaining bars, which correspond to PFASST(8, 7) on two, four and eight processors. Hence, if the workload of PFASST (8, 7) is distributed across sufficiently many processors, then the total runtime becomes smaller than the time-to-solution of the serial SDC (12,7) method. This is highlighted by (12, 7)) and PFASST (8,7, Z) with Z = 1, 2, 4, 8 time-processes (abbrev.…”

“…To solve the initial value problem (1), classical explicit time-stepping algorithms such as fourth-order RungeKutta schemes are commonly used, see e. g. [19]. Here, we use SDC methods [6], which can easily produce high-order time integration schemes from simple loworder methods and which can be parallelized in time using PFASST [7,17].…”

“…Furthermore, PFASST employs Full Approximation Scheme (FAS) corrections to increase the accuracy of SDC iterations on coarse levels. Many details of SDC, Parareal and PFASST have been omitted here for brevity, and the reader is referred to the more detailed discussions in e. g. [6,7,15,17,20,26].…”

“…The drawback of Parareal is that the parallel efficiency is formally bounded by 1/K where K is the number of iterations required for convergence. The Parallel Full Approximation Scheme in Space and Time (PFASST) method for parallelizing SDC methods in time is introduced in [7,17]. By combining the iterations of SDC with the iterations of Parareal, it significantly relaxes the efficiency bound of Parareal and further enhances the competitiveness of integration schemes based on SDC.…”