Efficient Implementation of Nonlinear Compact Schemes on Massively Parallel Platforms

Abstract

Weighted nonlinear compact schemes are ideal for simulating compressible, turbulent flows because of their nonoscillatory nature and high spectral resolution. However, they require the solution to banded systems of equations at each time-integration step or stage. We focus on tridiagonal compact schemes in this paper. We propose an efficient implementation of such schemes on massively parallel computing platforms through an iterative substructuring algorithm to solve the tridiagonal system of equations. The key features of our implementation are that it does not introduce any parallelization-based approximations or errors and it involves minimal neighbor-toneighbor communications. We demonstrate the performance and scalability of our approach on the IBM Blue Gene/Q platform and show that the compact schemes are efficient and have performance comparable to that of standard noncompact finite-difference methods on large numbers of processors (∼ 500, 000) and small subdomain sizes (four points per dimension per processor). Introduction.Weighted, nonlinear compact schemes use the adaptive stencil selection of the weighted, essentially nonoscillatory (WENO) [29,46] schemes to yield essentially nonoscillatory solutions with high spectral resolution; they are thus ideal for simulating compressible, turbulent flows. Notable efforts include weighted compact nonlinear schemes (WCNSs) [13,14,52,50], hybrid compact-ENO/WENO schemes [6,5,38,42], weighted compact schemes (WCSs) [30,33,51], compact-reconstruction WENO (CRWENO) schemes [19,22,18,20], and finite-volume compact-WENO (FVCW) schemes [25]. These schemes show a significant improvement in the resolution of moderate-and small-length scales compared with the resolution of the standard WENO schemes of the same (or higher) order and were applied to the simulation of compressible, turbulent flows. WCNSs [14,52,50] result in a system of equations with a linear left-hand side that can be prefactored. This is a substantial advantage; however, the spectral resolution of these schemes is only marginally higher than that of the WENO scheme. The hybrid compact-WENO, WCS, CRWENO, and FVCW schemes have a significantly higher spectral resolution, as demonstrated by both linear and nonlinear spectral analyses [38,20]. They result in solution-dependent systems of equations at each time-integration step or stage. Tests have shown that on

A fifth-order finite volume weighted compact scheme for solving one-dimensional Burgers’ equation

“…The stability condition for the present scheme with the fourth-order Runge-Kutta method is σ < 1.0 [27] and third-order SSP Runge-Kutta method (12) is σ < 0.9 [42].…”

Well-Balanced, Conservative Finite Difference Algorithm for Atmospheric Flows

“…Equation (33) results in a tridiagonal system of equations that must be solved at each timeintegration step or stage; however, the additional expense is justified by the higher accuracy and spectral resolution of the compact scheme [31,33,34]. An efficient and scalable implementation of the CRWENO5 scheme was recently proposed in [35,50] and is used in this study.…”

“…The CRWENO schemes have been applied to turbulent flows [33] and aerodynamic flows [34] where the resolution of small length scales is crucial. Although the CRWENO schemes require the solution to banded systems of equations at every timeintegration step or stage, a scalable implementation of the CRWENO scheme [35] demonstrated its performance for massively parallel simulations. The accuracy, spectral resolution, and scalability of the WENO and CRWENO schemes make them well suited for the simulation of atmospheric flows.…”

Well-Balanced Formulation of Gravitational Source Terms for Conservative Finite-Difference Atmospheric Flow Solvers

“…Equation (25) results in a tridiagonal system of equations that must be solved at each time-integration step or stage. An efficient and scalable implementation of the CRWENO5 scheme is proposed in 32,44 and used in this study. The solution of a hyperbolic system is composed of waves propagating at their characteristic speeds along their characteristic directions.…”