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A new parallel strategy for two-dimensional incompressible flow simulations using pseudo-spectral methods. (English) Zbl 1154.76361
Summary: A novel parallel technique for Fourier-Galerkin pseudo-spectral methods with applications to two-dimensional Navier-Stokes equations and inviscid Boussinesq approximation equations is presented. It takes advantage of the programming structure of the phase-shift de-aliased scheme for pseudo-spectral codes, and combines the task-distribution strategy [Z. Yin, H.J.H. Clercx, D.C. Montgomery, An easily implemented task-based parallel scheme for the Fourier pseudo-spectral solver applied to 2D Navier-Stokes turbulence, Comput. Fluid 33, 509 (2004)] and parallelized Fast Fourier Transform scheme. The performances of the resulting MPI Fortran90 codes with the new procedure on SGI 3800 are reported. For fixed resolution of the same problem, the peak speed of the new scheme can be twice as fast as the old parallel methods. The parallelized codes are used to solve some challenging numerical problems governed by the Navier-Stokes equations and the Boussinesq equations. Two interesting physical problems, namely, the double-valued ?-? structure in two-dimensional decaying turbulence and the collapse of the bubble cap in the Boussinesq simulation, are solved by using the proposed parallel algorithms.

MSC:
76M22 Spectral methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] Yin, Z.; Clercx, H.J.H.; Montgomery, D.C., An easily implemented task-based parallel scheme for the Fourier pseudospectral solver applied to 2D Navier-Stokes turbulence, Comput. fluid, 33, 509, (2004) · Zbl 1093.76048
[2] Orszag, S.A., Accurate solution of the Orr-Sommerfeld stability equation, J. fluid mech., 50, 689, (1971) · Zbl 0237.76027
[3] Patterson, G.S.; Orszag, S.A., Spectral calculations of isotropic turbulence: efficient removal of aliasing interaction, Phys. fluid, 14, 2538, (1971) · Zbl 0225.76033
[4] Canuto, C.; Hussaini, M.Y.; Quarteroni, A.; Zang, T.A., Spectral methods in fluid dynamics, (1987), Springer New York, Berlin · Zbl 0636.76009
[5] Boyd, J.P., Chebyshev and Fourier spectral methods, (2001), Dover Mineola/New York · Zbl 0987.65122
[6] Chen, S.; Doolen, G.D.; Kraichnan, R.H.; She, Z., On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. fluid A, 5, 458, (1993)
[7] Kaneda, Y.; Ishihara, T.; Yokokawa, M.; Itakura, K.; Uno, A., Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box, Phys. fluid, 15, L21, (2003)
[8] Moin, P.; Mahesh, K., Direct numerical simulation: a tool in turbulence research, Annu. rev. fluid mech., 30, 539, (1998) · Zbl 1398.76073
[9] Pelz, R.B., The parallel Fourier pseudospectral method, J. comput. phys., 92, 296, (1991) · Zbl 0709.76105
[10] Jackson, E.; She, Z.; Orszag, S.A., A case study in parallel computing: I. homogeneous turbulence on a hypercube, J. sci. comput., 6, 27, (1991) · Zbl 0737.76056
[11] Fischer, P.F.; Patera, A.T., Parallel simulation of viscous incompressible flows, Annu. rev. fluid mech., 26, 483, (1994) · Zbl 0802.76065
[12] Briscolini, M., A parallel implementation of a 3-D pseudospectral based code on the IBM 9076 scalable POWER parallel system, Parallel comput., 21, 1849, (1995)
[13] Dmitruk, P.; Wang, L.P.; Matthaeus, W.H.; Zhang, R.; Seckel, D., Scalable parallel FFT for spectral simulations on a beowulf cluster, Parallel comput., 27, 1921, (2001) · Zbl 0983.68234
[14] Iovieno, M.; Cavazzoni, C.; Tordella, D., A new technique for a parallel dealiased pseudospectral Navier-Stokes code, Comput. phys. commun., 141, 365, (2001) · Zbl 1041.76055
[15] Basu, A.J., A parallel algorithm for spectral solution of the three-dimensional Navier-Stokes equations, Parallel comput., 20, 1191, (1994) · Zbl 0817.76054
[16] Ling, W.; Liu, J.; Chung, J.N.; Crowe, C.T., Parallel algorithms for particles-turbulence two-way interaction direct numerical simulation, J. parallel distributed comput., 62, 38, (2002) · Zbl 1007.68208
[17] Fournier, E.; Gauthier, S.; Renaud, F., 2D pseudo-spectral parallel Navier-Stokes simulations of compressible Rayleigh-Taylor instability, Comput. fluid, 31, 569, (2002) · Zbl 1059.76053
[18] Bracco, A.; McWilliams, J.C.; Murante, G.; Provenzale, A.; Weiss, J.B., Revisiting freely decaying two-dimensional turbulence at millennial resolution, Phys. fluid, 12, 2931, (2000) · Zbl 1184.76069
[19] Dmitruk, P.; Montgomery, D.C., Numerical study of the decay of enstrophy in a two-dimensional Navier-Stokes fluid in the limit of very small viscosities, Phys. fluid, 17, 035114, (2005) · Zbl 1187.76128
[20] Yin, Z.; Montgomery, D.C.; Clercx, H.J.H., Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of “patches“ and “points”, Phys. fluid, 15, 1937, (2003) · Zbl 1186.76590
[21] Yin, Z., On final states of two-dimensional decaying turbulence, Phys. fluid, 16, 4623, (2004) · Zbl 1187.76584
[22] Pumir, A.; Siggia, E.D., Development of singular solutions to the axisymmetric Euler equations, Phys. fluid A, 4, 1472, (1992) · Zbl 0825.76121
[23] E, W.; Shu, C., Small-scale structures in Boussinesq convection, Phys. fluid, 6, 49, (1994) · Zbl 0822.76087
[24] Ceniceros, H.D.; Hou, T.Y., An efficient dynamically adaptive mesh for potentially singular solutions, J. comput. phys., 172, 609, (2001) · Zbl 0986.65087
[25] Montgomery, D.; Joyce, G.R., Statistical mechanics of negative temperature states, Phys. fluid, 17, 1139, (1974)
[26] Montgomery, D.; Matthaeus, W.H.; Stribling, W.T.; Martinez, D.; Oughton, S., Relaxation in two dimensions and the “sinh-poisson” equation, Phys. fluid A, 4, 3, (1992) · Zbl 0850.76485
[27] Orszag, S.A., Numerical simulation of incompressible flows within simple boundaries. I. Galerkin (spectral) representation, Stud. appl. math., 50, 293, (1971) · Zbl 0237.76012
[28] Swarztrauber, P.N., Multiprocessor ffts, Parallel comput., 5, 197, (1987) · Zbl 0624.65146
[29] Chamberlain, R.M., Gray codes, fast Fourier transforms and hypercubes, Parallel comput., 6, 225, (1988) · Zbl 0635.65145
[30] Pelz, R.B., Parallel compact FFTs for real sequences, SIAM J. sci. comput., 14, 914, (1993) · Zbl 0784.65108
[31] Mermer, C.; Kim, D.; Kim, Y., Efficient 2D FFT implementation on mediaprocessors, Parallel comput., 29, 691, (2003) · Zbl 1243.65167
[32] Chu, E.; George, A., Inside the FFT black box, (2000), CRC Press Boca Raton, London, New York, Washington, DC
[33] Chu, C.Y., Comparison of two-dimensional FFT methods on the hypercube, (), 1430
[34] Foster, I., Designing and building parallel programs: concepts and tools for parallel software engineering, (1995), Addison Wesley, Pearson Education, Inc. · Zbl 0844.68040
[35] D. Roose, R. Van Driessche, Parallel computers and parallel algorithms for CFD: an introduction, Special Course on Parallel Computing in CFD, AGARD R-807, NATO, 1995, p. 1.1.
[36] Fraguela, B.B.; Doallo, R.; Touriño, J.; Zapata, E.L., A compiler tool to predict memory hierarchy performance of scientific codes, Parallel comput., 30, 225, (2004), (and references therein)
[37] Culler, D.E.; Singh, J.P.; Gupta, A., Parallel computer architecture, (1999), Morgan Kaufmann Publishers, Inc. San Francisco
[38] Ghosh, S.; Hossain, M.; Matthaeus, W.H., The application of spectral methods in simulating compressible fluid and magnetofluid turbulence, Comput. phys. commun., 74, 18, (1993) · Zbl 0875.76407
[39] Clercx, H.J.H., A spectral solver for the Navier-Stokes equations in the velocity-vorticity formulation for flows with two nonperiodic directions, J. comput. phys., 137, 186, (1997) · Zbl 0904.76058
[40] Molenaar, D.; Clercx, H.J.H.; van Heijst, G.J.F., Angular momentum of forced 2D turbulence in a square no-slip domain, Physica D, 196, 329, (2004) · Zbl 1098.76546
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