PoisFFT – a free parallel fast Poisson solver. (English) Zbl 1410.65417

Summary: A fast Poisson solver software package PoisFFT is presented. It is available as a free software licensed under the GNU GPL license version 3. The package uses the fast Fourier transform to directly solve the Poisson equation on a uniform orthogonal grid. It can solve the pseudo-spectral approximation and the second order finite difference approximation of the continuous solution. The paper reviews the mathematical methods for the fast Poisson solver and discusses the software implementation and parallelization. The use of PoisFFT in an incompressible flow solver is also demonstrated.


65N06 Finite difference methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
65Y15 Packaged methods for numerical algorithms
Full Text: DOI arXiv


[1] Brown, D. L.; Cortez, R.; Minion, M. L., Accurate projection methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 168, 2, 464-499, (2001) · Zbl 1153.76339
[2] Hockney, R. W., A fast direct solution of poisson’s equation using Fourier analysis, J. ACM (JACM), 12, 1, 95-113, (1965) · Zbl 0139.10902
[3] Buzbee, B. L.; Golub, G. H.; Nielson, C. W., On direct methods for solving poisson’s equations, SIAM J. Numer. Anal., 7, 4, 627-656, (1970) · Zbl 0217.52902
[4] Buneman, O., Analytic inversion of the five-point Poisson operator, J. Comput. Phys., 8, 3, 500-505, (1971), ISSN 0021-9991, URL http://www.sciencedirect.com/science/article/pii/0021999171900295 · Zbl 0226.65066
[5] Cooley, J.; Lewis, P.; Welch, P., The fast Fourier transform algorithm: programming considerations in the calculation of sine, cosine and Laplace transforms, J. Sound Vib., 12, 3, 315-337, (1970) · Zbl 0195.46301
[6] Wilhelmson, R. B.; Ericksen, J. H., Direct solutions for poisson’s equation in three dimensions, J. Comput. Phys., 25, 4, 319-331, (1977) · Zbl 0373.65052
[7] Swarztrauber, P., The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of poisson’s equation on a rectangle, SIAM Rev., 19, 3, 490-501, (1977) · Zbl 0358.65088
[8] Swarztrauber, P. N., Symmetric ffts, Math. Comput., 47, 175, 323-346, (1986) · Zbl 0615.42008
[9] Schumann, U.; Sweet, R. A., Fast Fourier transforms for direct solution of poisson’s equation with staggered boundary conditions, J. Comput. Phys., 75, 1, 123-137, (1988) · Zbl 0642.65070
[10] Gockenbach, M. S., Understanding and implementing the finite element method, (2006), SIAM · Zbl 1105.65112
[11] Laizet, S.; Li, N., Incompact3d: a powerful tool to tackle turbulence problems with up to O (105) computational cores, Int. J. Numer. Methods Fluids, 67, 11, 1735-1757, (2011) · Zbl 1419.76481
[12] García-Risueño, P.; Alberdi-Rodriguez, J.; Oliveira, M. J.T.; Andrade, X.; Pippig, M.; Muguerza, J.; Arruabarrena, A.; Rubio, A., A survey of the parallel performance and accuracy of Poisson solvers for electronic structure calculations, J. Comput. Chem., 35, 6, 427-444, (2014)
[13] Swarztrauber, P. N.; Sweet, R. A., Algorithm 541: efficient Fortran subprograms for the solution of separable elliptic partial differential equations [D3], ACM Trans. Math. Software (TOMS), 5, 3, 352-364, (1979) · Zbl 0419.35043
[14] Frigo, M.; Johnson, S., The design and implementation of FFTW3, Proc. IEEE, 93, 2, 216-231, (2005), ISSN 0018-9219, URL http://www.bibsonomy.org/bibtex/2744e75c05789f40e27a64bd7d40462df/jpowell
[15] Pippig, M., PFFT: an extension of FFTW to massively parallel architectures, SIAMJSC, 35, 03, C213-C236, (2013) · Zbl 1275.65098
[16] Fuka, V.; Brechler, J., Large eddy simulation of the stable boundary layer, (Fořt, J.; Fürst, J.; Halama, J.; Herbin, R.; Hubert, F., Finite Volumes for Complex Applications VI Problems & Perspectives, Springer Proceedings in Mathematics, vol. 4, (2011), Springer Berlin, Heidelberg), 485-493 · Zbl 1246.76091
[17] Nicoud, F.; Toda, H. B.; Cabrit, O.; Bose, S.; Lee, J., Using singular values to build a subgrid-scale model for large eddy simulations, Phys. Fluids, 23, 8, 085106, (2011)
[18] Spalart, P. R.; Moser, R. D.; Rogers, M. M., Spectral methods for the Navier-Stokes equations with one infinite and two periodic directions, J. Comput. Phys., 96, 2, 297-324, (1991) · Zbl 0726.76074
[19] Kim, J.; Kim, D.; Choi, H., An immersed-boundary finite-volume method for simulations of flow in complex geometries, J. Comput. Phys., 171, 1, 132-150, (2001), ISSN 0021-9991, URL http://www.sciencedirect.com/science/article/pii/S0021999101967786 · Zbl 1057.76039
[20] Mittal, R.; Iaccarino, G., Immersed boundary methods, Annu. Rev. Fluid Mech., 37, 1, 239-261, (2005) · Zbl 1117.76049
[21] Peller, N.; Duc, A. L.; Tremblay, F.; Manhart, M., High-order stable interpolations for immersed boundary methods, Int. J. Numer. Methods Fluids, 52, 11, 1175-1193, (2006) · Zbl 1149.76633
[22] R. Fischer, I. Bastigkeit, B. Leitl, M. Schatzmann, Generation of spatio-temporally high resolved datasets for the validation of LES-models simulating flow and dispersion phenomena within the lower atmospheric boundary layer, in: Proceedings of Computational Wind Engineering 2010, 2010, <ftp://ftp.atdd.noaa.gov/pub/cwe2010/Files/Papers/461Fischer.pdf>.
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