×

Analysis of an explicit and matrix-free fractional step method for incompressible flows. (English) Zbl 1123.76054

Summary: We present an analysis of explicit and matrix-free fractional step method for incompressible flows. The presented method can be employed in either conservative or non-conservative form. The stabilization, convergence and conservation aspects of the presented method are discussed. A procedure for eliminating the first-order error in time introduced by the split is proposed. Some benchmark steady and unsteady state examples demonstrate the new aspects of the matrix-free scheme.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Zienkiewicz, O.C.; Taylor, R.L.; Nithiarasu, P., The finite element method for fluid dynamics, (2005), Elsevier Amsterdam
[2] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/petrov – galerkin formulation for convection dominated flows with particular emphasis on the incompressible navier – stokes equation, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[3] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M., A new finite element formulation for computational fluid dynamics: VIII. the Galerkin/least-squares method for advective diffusive equations, Comput. methods appl. mech. engrg., 73, 173-189, (1989) · Zbl 0697.76100
[4] Onate, E., Derivation of stabilized equations for numerical solution of advective – diffusive transport and fluid flow problems, Comput. methods appl. mech. engrg., 151, 233-265, (1998) · Zbl 0916.76060
[5] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid Sale models, bubbles and the origins of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[6] Codina, R., Comparison of some finite element methods for solving the diffusion – convection – reaction equations, Comput. methods appl. mech. engrg., 156, 185-210, (1998) · Zbl 0959.76040
[7] Löhner, R.; Morgan, K.; Zienkiewicz, O.C., The solution of non-linear hyperbolic equation systems by the finite element method, Int. J. numer. methods fluids, 4, 1043-1063, (1984) · Zbl 0551.76002
[8] Donea, J.; Huerta, A., Finite element methods for flow problems, (2003), Wiley London
[9] Douglas, J.; Russel, T.F., Numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. numer. anal., 19, 871-875, (1982) · Zbl 0492.65051
[10] Pironneau, O., On the transport diffusion algorithm and its application to the navier – stokes equation, Numer. math., 38, 309-332, (1982) · Zbl 0505.76100
[11] Morton, K.W., Generalised Galerkin methods for hyperbolic problems, Comput. methods appl. mech. engrg., 52, 847-871, (1985) · Zbl 0568.76007
[12] Morgan, K.; Peraire, J., Unstructured grid finite element methods for fluid mechanics, Rep. prog. phys., 61, 569-638, (1998)
[13] Zienkiewicz, O.C.; Codina, R., A general algorithm for compressible and incompressible flow—part I: the split, characteristic-based scheme, Int. J. numer. methods fluids, 20, 869-885, (1995) · Zbl 0837.76043
[14] Zienkiewicz, O.C.; Nithiarasu, P.; Codina, R.; Vázquez, M.; Ortiz, P., The characteristic based split procedure: an efficient and accurate algorithm for fluid problems, Int. J. numer. methods fluids, 31, 359-392, (1999) · Zbl 0985.76069
[15] Nithiarasu, P.; Codina, R.; Zienkiewicz, O.C., Characteristic based split (CBS) scheme A unified approach to fluid dynamics, Int. J. numer. methods engrg., (2006), (special issue) · Zbl 1110.76324
[16] Ewing, R.E.; Russel, T.F., Multistep Galerkin methods along characteristics for convection – diffusion problems, (), 28-36
[17] Bercovier, M.; Pironneau, O.; Sastri, V., Finite elements and characteristics for some parabolic-hyperbolic problems, Appl. methods model., 7, 89-96, (1983) · Zbl 0505.65055
[18] Pironneau, O.; Liou, J.; Tezduyar, T.T.I., Chractersitc gaerlin and Galerkin least-squares space-time formulations for the advection – diffusion equation with time dependent domain, Comp. math. appl. mech. engrg., 100, 117-141, (1992) · Zbl 0761.76073
[19] Drikakis, D.; Govatsos, P.; Papantonis, D., A characteristic based method for incompressible flow, Int. J. numer. methods fluids, 19, 667-685, (1994) · Zbl 0817.76058
[20] Kaazempur-Mofrad, M.R.; Ethier, C.R., An efficient characteristic Galerkin scheme for the advection equation in 3-D, Comput. methods appl. mech. engrg., 191, 5345-5363, (2002) · Zbl 1083.76552
[21] Kaazempur-Mofrad, M.R.; Minev, P.D.; Ethier, C.R., A characteristic/finite element algorithm for time-dependent 3-D advection dominated transport using unstructured grids, Comput. methods appl. mech. engrg., 192, 1281-1298, (2003) · Zbl 1059.76035
[22] Drikakis, D.; Rider, W., High-resolution methods for incompressible and low speed flows, (2004), Springer Berlin
[23] Chorin, A.J., Numerical solution of navier – stokes equations, Math. comput., 22, 745-762, (1968) · Zbl 0198.50103
[24] Nithiarasu, P., An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows, Int. J. numer. methods engrg., 56, 1815-1845, (2003) · Zbl 1072.76040
[25] Nithiarasu, P.; Mathur, J.S.; Weatherill, N.P.; Morgan, K., Three-dimensional incompressible flow calculations using the characteristic based split (CBS) scheme, Int. J. numer. methods fluids, 44, 1207-1229, (2004) · Zbl 1067.76572
[26] Nithiarasu, P.; Liu, C.-B., Steady and unstable flow calculations in a double driven cavity using the explicit CBS scheme, Int. J. numer. methods engrg., 63, 380-397, (2005) · Zbl 1140.76397
[27] P. Nithiarasu, C.-B. Liu, An explicit characteristic based split (CBS) scheme for incompressible turbulent flows, Comput. Methods Appl. Mech. Engrg., in press, doi:10.1016/j.cma.2004.09.017. · Zbl 1176.76086
[28] Nithiarasu, P.; Massarotti, N.; Mathur, J.S., Forced convection heat transfer from solder balls on a printed circuit board using the characteristic based split (CBS) scheme, Int. J. numer. methods heat fluid flow, 15, 73-95, (2005) · Zbl 1162.76392
[29] Nithiarasu, P., An arbitrary eulerian Lagrangian (ALE) method for free surface flow calculations using the characteristic based split (CBS) scheme, Int. J. numer. methods fluids, 48, 1415-1428, (2005) · Zbl 1072.76041
[30] Nithiarasu, P., On boundary conditions of the CBS algorithm for computational fluid dynamics, Int. J. numer. methods engrg., 54, 523-536, (2002) · Zbl 1008.76040
[31] Zienkiewicz, O.C.; Sai, B.V.K.S.; Morgan, K.; Codina, R.; Vázquez, M., A general algorithm for compressible and incompressible flow—part II. tests on the explicit form, Int. J. numer. methods fluids, 20, 887-913, (1995) · Zbl 0837.76044
[32] Codina, R.; Vázquez, M.; Zienkiewicz, O.C., General algorithm for compressible and incompressible flows, part III—A semi-implicit form, Int. J. numer. methods fluids, 27, 13-32, (1998) · Zbl 0914.76050
[33] Zienkiewicz, O.C.; Rojek, J.; Taylor, R.L.; Pastor, M., Triangles and tetrahedra in explicit dynamic codes for solids, Int. J. numer. methods engrg., 43, 565-583, (1999) · Zbl 0939.74073
[34] Zienkiewicz, O.C.; Ortiz, P., A split characteristic based finite element model for shallow water equations, Int. J. numer. methods fluids, 20, 1061-1080, (1995) · Zbl 0836.76052
[35] Salomoni, V.A.; Schrefler, B.A., A CBS-type stabilizing algorithm for the consolidation of saturated porous media, Int. J. numer. methods engrg., 63, 502-527, (2005) · Zbl 1140.76400
[36] Thomas, C.G.; Nithiarasu, P., Effect of variable smoothing and stream line direction on the viscous compressible flow calculations, Int. J. numer. methods heat fluid flow, 15, 420-428, (2005) · Zbl 1231.76237
[37] Codina, R.; Zienkiewicz, O.C., CBS versus GLS stabilisation of the incompressible navier – stokes equations and the role of the time step as stabilisation parameter, Commun. numer. methods engrg., 18, 99-112, (2002) · Zbl 1093.76528
[38] Codina, R.; Coppola-Owen, H.; Nithiarasu, P.; Liu, C.-B., Numerical comparison of CBS and SGS as stabilisation techniques for the incompressible navier – stokes equations, Int. J. numer. methods engrg., (2006), (special issue) · Zbl 1110.76312
[39] Rannacher, R., The navier – stokes equations II—theory and numerical methods, Lecture notes in mathematics, vol. 1530, (1992), Springer-Verlag Berlin
[40] Strikwerda, J.C.; Lee, Y.S., The accuracy of the fractional step method, SIAM J. numer. anal., 37, 37, (1999) · Zbl 0953.65061
[41] Brown, D.; Cortez, R.; Minion, M., Accurate projection methods for the incompressible navier – stokes equations, J. comput. phys., 168, 464, (2001) · Zbl 1153.76339
[42] Chang, W.; Giraldo, F.; Perot, B., Analysis of an exact fractional step method, J. comput. phys., 18, 183-199, (2002) · Zbl 1130.76394
[43] Codina, R., Pressure stability in fractional step finite element methods for incompressible flows, J. comput. phys., 170, 112-140, (2001) · Zbl 1002.76063
[44] Malan, A.G.; Lewis, R.W.; Nithiarasu, P., An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: part I. theory and implementation, Int. J. numer. methods engrg., 54, 695-714, (2002) · Zbl 1098.76581
[45] Gaitonde, A.L., A dual time method for two dimensional incompressible flow calculations, Int. J. numer. methods egnrg., 41, 1153-1166, (1998) · Zbl 0911.76044
[46] J. Abanto, D. Pelletier, A. Gadon, J.-Y. Trepanier, M. Reggio, Verification of some commercial CFD codes on atypical CFD problems, AIAA Paper 2005-0682, 10-13 January 2005, Reno.
[47] Dennis, S.C.R.; Chang, G.Z., Numerical solutions for steady flow past a circular cylinder at Reynolds numbers upto 100, J. fluid mech., 42, 471, (1970) · Zbl 0193.26202
[48] Takami, H.; Keller, H.B., Steady two-dimensional viscous flow of an incompressible fluid past a circular cylinder, Phys. fluids, 12, II-51, (1969) · Zbl 0206.55004
[49] Tuann, S.Y.; Olson, M.D., Numerical studies of the flow around a circular cylinder by a finite element method, Comput. fluids, 6, 219, (1978) · Zbl 0394.76038
[50] Ding, H.; Shu, C.; Yeo, K.S.; Xu, D., Simulation of incompressible viscous flow past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method, Comput. methods appl. mech. engrg., 193, 727-744, (2004) · Zbl 1068.76062
[51] Ghia, U.; Ghia, K.N.; Chin, C.T., High re solutions for incompressible flow using the navier – stokes equation and multigrid methods, J. comput. phys., 48, 387, (1982) · Zbl 0511.76031
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.