Efficient parallel computing with a compact finite difference scheme. (English) Zbl 1365.65196

Kim, Jae Wook; Sandberg, Richard D.

Comput. Fluids 58, 70-87 (2012).

Summary: This paper proposes an efficient parallel computing approach based on a high-order accurate compact finite difference scheme in conjunction with a conventional domain decomposition method and MPI libraries. The proposed parallel computing approach consists of two major features: (a) a newly developed compact finite difference scheme with extended stencils containing halo points around subdomain boundaries, and (b) a predictor-corrector type implementation of a compact filter that effectively suppresses spurious errors from the subdomain boundaries. The current work employs three halo cells for the inter-node communication, based on which the coefficients of the new compact scheme at the subdomain boundaries are optimized to achieve as high level of resolution and accuracy as the interior compact scheme provides. Also, an optimal set of cut-off wavenumbers of the compact filter that minimizes spurious errors is suggested. It is shown that the level of errors from the proposed parallel calculations lies within the same order of magnitude of that from the single-domain serial calculations. The overall accuracy and linear stability of the new parallel compact differencing-filtering system are confirmed by grid convergence tests and eigenvalue analyses. The proposed approach shows a substantial improvement with respect to existing methods available.

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65Y05	Parallel numerical computation
76M20	Finite difference methods applied to problems in fluid mechanics

Keywords:

parallel computing; compact scheme; high order; halo cell

Software:

MPI

References:

[1]	Laizet, S.; Lamballais, E., High-order compact schemes for incompressible flows: a simple and efficient method with quasi-spectral accuracy, J comput phys, 228, 16, 5989, (2009) · Zbl 1185.76823
[2]	Schaupp, C.; Sesterhenn, J.; Friedrich, R., On a method for direct numerical simulation of shear layer/compression wave interaction for aeroacoustics investigations, Comput fluids, 37, 4, 463, (2008) · Zbl 1237.76108
[3]	Bodony, D.; Lele, S.K., Current status of jet noise predictions using large-eddy simulation, Aiaa j, 46, 2, 364, (2008)
[4]	Cook, A.W., Artificial fluid properties for large-eddy simulation of compressible turbulent mixing, Phys fluids, 19, 055103, (2007) · Zbl 1146.76358
[5]	Kim, J.W., Optimised boundary compact finite difference schemes for computational aeroacoustics, J comput phys, 225, 1, 995, (2007) · Zbl 1118.76045
[6]	Sun XH, Moitra S. A fast parallel tridiagonal algorithm for a class of CFD applications. NASA technical paper 3585; 1996.
[7]	Hofhaus, J.; Van de Velde, E.F., Alternating-direction line-relaxation methods on multicomputers, SIAM J sci comput, 17, 2, 454, (1996) · Zbl 0851.65065
[8]	Povitsky, A.; Morris, P.J., A higher-order compact method in space and time based on parallel implementation of the Thomas algorithm, J comput phys, 161, 1, 182, (2000) · Zbl 0959.65102
[9]	Sengupta, T.K.; Dipankar, A.; Rao, A.K., A new compact scheme for parallel computing using domain decomposition, J comput phys, 220, 2, 654, (2007) · Zbl 1370.76072
[10]	Kim, J.W., High-order compact filters with variable cut-off wavenumber and stable boundary treatment, Comput fluids, 39, 7, 1168, (2010) · Zbl 1242.76204
[11]	Lele, S.K., Compact finite difference schemes with spectral-like resolution, J comput phys, 103, 1, 16, (1992) · Zbl 0759.65006
[12]	Carpenter, M.H.; Gottlieb, D.; Abarbanel, S., The stability of numerical boundary treatments for compact high-order finite-difference schemes, J comput phys, 108, 2, 272, (1993) · Zbl 0791.76052
[13]	Bogey, C.; Bailly, C., On the application of explicit spatial filtering to the variables or fluxes of linear equations, J comput phys, 255, 2, 1211, (2007) · Zbl 1122.65069
[14]	Gaitonde, D.V.; Visbal, M.R., Padé-type higher-order boundary filters for the navier – stokes equations, Aiaa j, 38, 11, 2103, (2000)
[15]	Kim, J.W.; Lee, D.J., Generalized characteristic boundary conditions for computational aeroacoustics, Aiaa j, 38, 11, 2040, (2000)
[16]	Sandberg, R.D., An axis treatment for flow equations in cylindrical coordinates based on parity conditions, Comput. fluids, 49, 1, 166, (2011) · Zbl 1271.76232
[17]	Kennedy, C.A.; Carpenter, M.H.; Lewis, R.M., Low-storage, explicit runge – kutta schemes for the compressible navier – stokes equations, Appl numer math, 35, 177, (2000) · Zbl 0986.76060
[18]	Kennedy, C.A.; Gruber, A., Reduced aliasing formulations of the convective terms within the navier – stokes equations for a compressible fluid, J comput phys, 227, 1676, (2008) · Zbl 1290.76135
[19]	Sandberg, R.D.; Sandham, N.D., Nonreflecting zonal characteristic boundary condition for direct numerical simulation of aerodynamic sound, Aiaa j, 44, 2, 402, (2006)
[20]	Fasel, H.F.; Konzelmann, U., Non-parallel stability of a flat-plate boundary layer using the complete navier – stokes equations, J fluid mech, 221, 311, (1990) · Zbl 0715.76019
[21]	Sandberg, R.D.; Sandham, N.D.; Joseph, P.F., Direct numerical simulations of trailing-edge noise generated by boundary-layer instabilities, J sound vib, 304, 3-5, 677, (2007)
[22]	Bogey, C.; Bailly, C., A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations, J comput phys, 228, 5, 1447, (2009) · Zbl 1263.76046