×

zbMATH — the first resource for mathematics

A Hermite pseudospectral solver for two-dimensional incompressible flows on infinite domains. (English) Zbl 1349.76576
Summary: The Hermite pseudospectral method is applied to solve the Navier-Stokes equations on a two-dimensional infinite domain. The feature of Hermite function allows us to adopt larger time steps than other spectral methods, but also leads to some extra computation when the stream function is calculated from the vorticity field. The scaling factor is employed to increase the resolution within the region of our main interest, and the aliasing error is fully removed by the 2/3-rule. Several traditional numerical experiments are performed with high accuracy, and some related future work on physical applications of this program is also discussed.

MSC:
76M22 Spectral methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
Software:
FHT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Montgomery, D.; Matthaeus, W., Oseen vortex as a maximum entropy state of a two dimensional fluid, Phys. Fluids, 23, 075104, (2011)
[2] Yin, Z.; Chang, L.; Hu, W.; Li, Q.; Wang, H., Numerical simulations on thermocapillary migrations of nondeformable droplets with large Marangoni numbers, Phys. Fluids, 24, 092101, (2012)
[3] Melander, M.; McWilliams, J.; Zabusky, N., Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation, J. Fluid Mech., 178, 137-159, (1987) · Zbl 0633.76023
[4] Platte, R.; Rossi, L.; Mitchell, T., Using global interpolation to evaluate the Biot-Savart integral for deformable elliptical Gaussian vortex elements, SIAM J. Sci. Comput., 31, 2342-2360, (2009) · Zbl 1192.65029
[5] Mariotti, A.; Legras, B.; Dritschel, D., Vortex stripping and the erosion of coherent structures in two-dimensional flows, Phys. Fluids, 6, 3954-3962, (1994)
[6] Shen, J.; Tang, T., Spectral and high-order methods with applications, (2006), Science Press Beijing · Zbl 1234.65005
[7] Shen, J.; Wang, L., Some recent advances on spectral methods for unbounded domains, Commun. Comput. Phys., 5, 195-241, (2009) · Zbl 1364.65265
[8] Boyd, J., Chebyshev and Fourier spectral methods, (2001), Dover New York · Zbl 0994.65128
[9] Tang, T., The Hermite spectral method for Gaussian-type functions, SIAM J. Sci. Comput., 14, 3, 594-606, (1993) · Zbl 0782.65110
[10] Funago, D.; Kavian, O., Approximation of some diffusion evolution equations in unbounded domains by Hermite functions, Math. Comput., 57, 196, 597-619, (1991) · Zbl 0764.35007
[11] Tang, T.; McKee, S.; Reeks, M., A spectral method for the numerical solutions of a kinetic equation describing the dispersion of small particles in a turbulent flow, J. Comput. Phys., 103, 222-230, (1992) · Zbl 0763.76064
[12] Guo, B., Error estimation of Hermite spectral method for nonlinear partial differential equations, Math. Comput., 68, 227, 1067-1078, (1999) · Zbl 0918.65069
[13] Guo, B.; Xu, C., Hermite pseudospectral method for nonlinear partial differential equations, Modél. Math. Anal. Numér., 34, 4, 859-872, (2000) · Zbl 0966.65072
[14] Fok, J.; Guo, B.; Tang, T., Combined Hermite spectral-finite difference method for the Fokker-Planck equation, Math. Comput., 71, 240, 1497-1528, (2001) · Zbl 1007.65068
[15] Schumer, J.; Holloway, J., Vlasov simulations using velocity-scaled Hermite representations, J. Comput. Phys., 144, 626-661, (1998) · Zbl 0936.76061
[16] Weishaupl, R.; Schmeiser, C.; Markowich, P.; Borgna, J., A Hermite pseudo-spectral method for solving systems of Gross-Pitaevskii equations, Commun. Math. Sci., 5, 2, 299-312, (2007) · Zbl 1151.35087
[17] Parand, K.; Dehghan, M.; Rezaei, A.; Ghaderi, S., An approximation algorithm for the solution of nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Commun., 181, 1096-1108, (2010) · Zbl 1216.65098
[18] Parand, K.; Rezaei, A.; Ghaderi, S., An approximate solution of MHD Falkner-Skan flow by Hermite functions pseudospectral method, Commun. Nonlinear Sci. Numer. Simul., 16, 274-283, (2011) · Zbl 1221.76131
[19] Hagedorn, G., Raising and lowering operators for semiclassical wave packets, Ann. Phys., 269, 77-104, (1998) · Zbl 0929.34067
[20] Kieri, E.; Holmgren, S.; Karlsson, H., An adaptive pseudospectral method for wave packet dynamics, J. Chem. Phys., 137, 044111, (2012)
[21] Xiang, X.; Wang, Z., Generalized Hermite approximations and spectral method for partial differential equations in multiple dimensions, J. Sci. Comput., 57, 229-253, (2013) · Zbl 1282.65161
[22] Driscoll, J.; Healy, D.; Rockmore, D., Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs, SIAM J. Comput., 26, 1066-1099, (1997) · Zbl 0896.65094
[23] Leibon, G.; Rockmore, D.; Park, W.; Taintor, R.; Chirikjian, G., A fast Hermite transform, Theor. Comput. Sci., 409, 211-228, (2008) · Zbl 1156.65104
[24] Press, W.; Teukolsky, S.; Vetterling, W.; Flannery, B., Numerical recipes in C++: the art of scientific computing, 55-58, (2002), Cambridge University Press
[25] Canuto, C.; Hussaini, M.; Quarteroni, A.; Zang, T., Spectral methods in fluid dynamics, 84-85, (1987), Springer-Verlag New York
[26] 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. Fluids, 33, 509-520, (2004) · Zbl 1093.76048
[27] Weideman, J., The eigenvalues of Hermite and rational spectral differentiation matrices, Numer. Math., 61, 409-432, (1992) · Zbl 0741.65078
[28] Robinson, A.; Saffman, P., Stability and structure of stretched vortices, Stud. Appl. Math., 70, 163-181, (1984) · Zbl 0549.76016
[29] Saffman, P.; Ablowitz, M.; Hinch, E.; Ockendon, J.; Olver, P., Vortex dynamics, (1992), Cambridge University Press Cambridge
[30] Koumoutsakos, P., Inviscid axisymmetrization of an elliptical vortex, J. Comput. Phys., 138, 821-857, (1997) · Zbl 0902.76080
[31] Yin, Z.; Gao, P.; Hu, W.; Chang, L., Thermocapillary migrations of nondeformable drops, Phys. Fluids, 20, 082101, (2008) · Zbl 1182.76854
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.