Explicit parallel-in-time integration of a linear acoustic-advection system

Abstract

The applicability of the Parareal parallel-in-time integration scheme for the solution of a linear, two-dimensional hyperbolic acoustic-advection system, which is often used as a test case for integration schemes for numerical weather prediction (NWP), is addressed. Parallel-in-time schemes are a possible way to increase, on the algorithmic level, the amount of parallelism, a requirement arising from the rapidly growing number of CPUs in high performance computer systems. A recently introduced modification of the "parallel implicit time-integration algorithm" could successfully solve hyperbolic problems arising in structural dynamics. It has later been cast into the framework of Parareal. The present paper adapts this modified Parareal and employs it for the solution of a hyperbolic flow problem, where the initial value problem solved in parallel arises from the spatial discretization of a partial differential equation by a finite difference method. It is demonstrated that the modified Parareal is stable and can produce reasonably accurate solutions while allowing for a noticeable reduction of the time-to-solution. The implementation relies on integration schemes already widely used in NWP (RK-3, partially split forward Euler, forward-backward). It is demonstrated that using an explicit partially split scheme for the coarse integrator allows to avoid the use of an implicit scheme while still achieving speedup.

On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

“…They applied this idea to the linear hyperbolic problems in structural dynamics. Recently, a similar idea was successfully applied to linear hyperbolic problems [21]. The basic idea of the Krylov subspace parareal method is to project u k+1 i onto a subspace spanned by all numerical solutions integrated by the fine solver at previous iterations.…”

“…The compression ratio is R = r/N . Following the notation of [21]: In [21], the speedup is estimated as…”

“…Another attempt, the Krylov subspace parareal method, builds a new coarse solver by reusing all information from the corresponding fine solver at previous iterations. The stability of this approach was demonstrated for linear problems on linear structural dynamics [10] and a linear 2-D acoustic-advection system [21]. However, the Krylov subspace parareal method appears to be limited to linear problems.…”

“…Inspired by [13,21], we propose a modified parareal method, referred to as the reduced basis parareal method in which the Krylov subspace is replaced by a subspace spanned by a set of reduced bases, constructed on-the-fly from the fine solver. This method inherits most advantages of the Krylov subspace parareal method and is observed to retain stability and convergence for linear wave problems.…”

Integrating an N-Body Problem with SDC and PFASST

“…A very general scheme is Parareal [15], which allows arbitrary integration schemes to be used in a black-box fashion. A detailed mathematical analysis of Parareal is conducted in [8] and comprehensive lists of references can be found e. g. in [17,20]. 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.…”