×

zbMATH — the first resource for mathematics

A splitting method for incompressible flows with variable density based on a pressure Poisson equation. (English) Zbl 1159.76028
Summary: We propose a new fractional time-stepping technique for solving incompressible flows with variable density. The main feature of this method is that, as opposed to other known algorithms, the pressure is determined by just solving one Poisson equation per time step, which greatly reduces the computational cost. The stability of the method is proved, and the performance of the method is numerically illustrated.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Almgren, Ann S.; Bell, John B.; Colella, Phillip; Howell, Louis H.; Welcome, Michael L., A conservative adaptive projection method for the variable density incompressible navier – stokes equations, J. comput. phys., 142, 1, 1-46, (1998) · Zbl 0933.76055
[2] Bell, John B.; Marcus, Daniel L., A second-order projection method for variable-density flows, J. comput. phys., 101, 334-348, (1992) · Zbl 0759.76045
[3] Brezzi, F.; Fortin, M., Mixed and hybrid finite element methods, (1991), Springer-Verlag New York, NY · Zbl 0788.73002
[4] 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
[5] Chorin, A.J., Numerical solution of the navier – stokes equations, Math. comput., 22, 745-762, (1968) · Zbl 0198.50103
[6] Douglas, J.; Russell, 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, 871-885, (1982) · Zbl 0492.65051
[7] Ern, A.; Guermond, J.-L., Theory and practice of finite elements, Applied mathematical sciences, vol. 159, (2004), Springer-Verlag New York
[8] Fraigneau, Y.; Guermond, J.-L.; Quartapelle, L., Approximation of variable density incompressible flows by means of finite elements and finite volumes, Commun. numer. methods eng., 17, 893-902, (2001) · Zbl 1022.76029
[9] Girault, V.; Raviart, P.-A., Finite element methods for navier – stokes equations theory and algorithms Springer series in computational mathematics, (1986), Springer-Verlag Berlin, Germany
[10] Guermond, J.-L., Some practical implementations of projection methods for navier – stokes equations, M2AN math. model. numer. anal., 30, 5, 637-667, (1996) · Zbl 0861.76065
[11] Guermond, J.-L., Un résultat de convergence d’ordre deux en temps pour l’approximation des équations de navier – stokes par une technique de projection incrémentale, M2AN math. model. numer. anal., 33, 1, 169-189, (1999), Also in C.R. Acad. Sci. Paris, Série I, 325:1329-1332, 1997 · Zbl 0921.76123
[12] Guermond, J.-L.; Marra, A.; Quartapelle, L., Subgrid stabilized projection method for 2d unsteady flows at high Reynolds number, Comput. methods appl. mech. eng., 195, (2006) · Zbl 1121.76036
[13] Guermond, J.L.; Minev, P.; Shen, Jie, An overview of projection methods for incompressible flows, Comput. methods appl. mech. eng., 195, 44-47, 6011-6045, (2006) · Zbl 1122.76072
[14] Guermond, J.-L.; Pasquetti, R., Entropy-based nonlinear viscosity for Fourier approximations of conservation laws, C.R. math. acad. sci. Paris, 346, 913-918, (2008)
[15] J.-L. Guermond, R. Pasquetti, B. Popov, Entropy-based viscosities, in preparation. · Zbl 1421.76039
[16] Guermond, J.-L.; Quartapelle, L., Calculation of incompressible viscous flows by an unconditionally stable projection FEM, J. comput. phys., 132, 1, 12-33, (1997) · Zbl 0879.76050
[17] Guermond, J.-L.; Quartapelle, L., On the approximation of the unsteady navier – stokes equations by finite element projection methods, Numer. math., 80, 5, 207-238, (1998) · Zbl 0914.76051
[18] Guermond, J.-L.; Quartapelle, L., A projection FEM for variable density incompressible flows, J. comput. phys., 165, 1, 167-188, (2000) · Zbl 0994.76051
[19] Guermond, J.-L.; Shen, J., On the error estimates for the rotational pressure-correction projection methods, Math. comput., 73, 248, 1719-1737, (2004), electronic · Zbl 1093.76050
[20] 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 advection-diffusive equations, Comput. methods appl. mech. eng., 73, 173-189, (1989) · Zbl 0697.76100
[21] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic equations, Comput. methods appl. mech. eng., 45, 285-312, (1984) · Zbl 0526.76087
[22] P.-L. Lions, Mathematical topics in fluid mechanics, vol. 1. Incompressible models, Oxford Lecture Series in Mathematics and its Applications, vol. 3. The Clarendon Press, Oxford University Press, New York, 1996.
[23] Liu, Chun; Walkington, Noel J., Convergence of numerical approximations of the incompressible navier – stokes equations with variable density and viscosity, SIAM J. numer. anal., 45, 3, 1287-1304, (2007), electronic · Zbl 1138.76048
[24] Pyo, Jae-Hong; Shen, Jie, Gauge-Uzawa methods for incompressible flows with variable density, J. comput. phys., 221, 1, 181-197, (2007) · Zbl 1109.76037
[25] R. Rannacher, On Chorin’s projection method for the incompressible Navier-Stokes equations, in: The Navier-Stokes Equations II—Theory and Numerical Methods (Oberwolfach, 1991), Lecture Notes in Math., vol. 1530, Springer, Berlin, Germany, 1992, pp. 167-183. · Zbl 0769.76053
[26] Shen, J., On error estimates of projection methods for the navier – stokes equations: first-order schemes, SIAM J. numer. anal., 29, 57-77, (1992) · Zbl 0741.76051
[27] J. Shen, Efficient Chebyshev-Legendre Galerkin methods for elliptic problems, in: A.V. Ilin, R.L. Scott (Eds.), Proceedings of ICOSAHOM’95, Houston J. Math., 1996, pp. 233-240.
[28] Temam, R., Sur l’approximation de la solution des équations de navier – stokes par la méthode des pas fractionnaires II, Arch. rat. mech. anal., 33, 377-385, (1969) · Zbl 0207.16904
[29] Temam, Roger, Une méthode d’approximation de la solution des équations de navier – stokes, Bull. soc. math. France, 96, 115-152, (1968) · Zbl 0181.18903
[30] Timmermans, L.J.P.; Minev, P.D.; van de Vosse, F.N., An approximate projection scheme for incompressible flow using spectral elements, Int. J. numer. methods fluids, 22, 673-688, (1996) · Zbl 0865.76070
[31] Tryggvason, G., Numerical simulation of rayleigh – taylor instability, J. comput. phys., 75, 253-282, (1988) · Zbl 0638.76056
[32] Walkington, N.J., Convergence of the discontinuous Galerkin method for discontinuous solutions, SIAM J. numer. anal., 42, 5, 1801-1817, (2004) · Zbl 1082.65088
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.