A High Accuracy Preserving Parallel Algorithm for Compact Schemes for DNS

Abstract

A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorithm with two benefits. First, the number of overlap points is reduced to a single common boundary point between any two neighboring sub-domains, thereby saving the number of points used, with resultant speed-up. Second, with a proper design, errors arising due to sub-domain boundary closure schemes are reduced to a user designed error tolerance, bringing the new parallel scheme on par with sequential computing. Error reduction is achieved by using global spectral analysis, introduced in “Analysis of central and upwind compact schemes,” J. Comput. Phys. 192, 2, (2003) 677--694, which analyzes any discrete computing method in the full domain integrally. The design of the parallel compact scheme is explained, followed by a demonstration of the accuracy of the method by solving benchmark flows: (1) periodic two-dimensional Taylor-Green vortex problem; (2) flow inside two-dimensional square lid-driven cavity (LDC) at high Reynolds number; and (3) flow inside a non-periodic three-dimensional cubic LDC with the staggered grid arrangement.

Ultrasound Triggering of Rayleigh-Taylor Instability: Solution of Compressible Navier-Stokes Equation by a Non-Overlapping Parallel Compact Scheme

“…An explicit scheme is to be used at the sub-domain boundaries in each processor to decouple the system of equations, which is the proposed sub-domain closure. The difference in the spectral resolution of the compact scheme used in the interior of the domain and the explicit scheme used at the sub-domain boundaries alters the numerical properties at a few grid points near the sub-domain boundaries [33]. This is the source of additional error in parallel computing of derivatives using compact schemes, a topic which has not been addressed except in [33].…”

“…The use of an explicit central scheme at the extended sub-domain boundaries reduces bias at the sub-domain boundary point. However, instead of increased accuracy of the closure schemes, discontinuous resolution at the sub-domain boundaries has been noted as a source of error [33]. To improve the efficiency of large-scale simulations, one would like to decompose the domain with fewer overlap points without compromising the accuracy by filtering the solution [34,47,48].…”

“…Moreover, the sub-domain closures present in [44,46,53] are specifically designed to work with a chosen compact scheme. In contrast, the NOHAP sub-domain closure [33] used in the present work does not require overlapping points at the sub-domain boundaries. Computationally this is equivalent to solving the sub-domain boundary point as equivalent to an interior point.…”

“…For any compact scheme for the first derivative, the equivalent explicit scheme can be obtained as [33],…”

“…Thus, the spatial resolution has improved significantly, and the time step has been refined to 7.69 × 10 −8 sec (with a non-dimensional time step of 6.25 × 10 −7 , as compared to a time step of 10 −6 in [12]. More importantly, a novel parallel algorithm (non-overlapping high accuracy parallel (NOHAP) compact scheme) proposed in [33] is used. The parallelization scheme ensures the removal of any error originating at the sub-domain boundaries due to boundary closure schemes mandatory for compact schemes.…”