Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows. (English) Zbl 1203.76122

Summary: We present a review of the semi-Lagrangian method for advection-diffusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] Loft, R. D., Thomas, S. J., and Dennis, J. M. (2001). Terascale spectral element dynamical core for atmospheric general circulation models. InProceedings of Supercomuting 2001, Denver.
[2] Maday, Y.; Patera, A. T.; Ronquist, E. M., An operator-integration-factor splitting method for time-dependent problems: Application to incompressible fluid flow, J. Sci. Comp., 4, 263-292 (1990) · Zbl 0724.76070
[3] Xiu, D.; Karniadakis, G. E., A semi-Lagrangian high-order method for Navier-Stokes equations, J. Comp. Phys., 172, 658-658 (2001) · Zbl 1028.76026
[4] Robert, A., A stable numerical integration scheme for the primitive meteorological equations, Atmos. Ocean, 19, 35-35 (1981)
[5] Pironneau, O., On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38, 309-309 (1982) · Zbl 0505.76100
[6] Giraldo, F. X., Strong and weak Lagrange-Galerkin spectral element methods for shallow water equations, Comput. Math. Appl., 45, 97-121 (2003) · Zbl 1029.76042
[7] Karniadakis, G. E., and Orszag, S. A. (1993). Nodes, modes, and flow codes.Phys. Today34.
[8] Karniadakis, G. E.; Sherwin, S. J., Spectral/hpElement Methods for CFD (1999), London: Oxford University Press, London · Zbl 0954.76001
[9] Falcone, M.; Ferretti, R., Convergence analysis for a class of high-order semi-Lagrangian advection schemes, SIAM J. Numer. Anal., 35, 909-909 (1998) · Zbl 0914.65097
[10] Huffenus, J. P.; Khaletzky, D., A finite element method to solve the Navier-Stokes equations using the method of characteristics, Int. J. Numer. Methods Fluids, 4, 247-247 (1984) · Zbl 0547.76038
[11] Malevsky, A. V.; Thomas, S. J., Parallel algorithms for semi-Lagrangian advection, Int. J. Numer. Methods Fluids, 25, 455-455 (1997) · Zbl 0910.76063
[12] Oliveira, A.; Baptista, A. M., A comparison of integration and interpolation Eulerian-Lagrangian methods, Int. J. Numer. Methods Fluids, 21, 183-183 (1995) · Zbl 0841.76041
[13] Staniforth, A.; Cote, J., Semi-Lagrangian integration schemes for atmospheric models — a review, Mon. Wea. Rev., 119, 2206-2206 (1991) · Zbl 0746.76054
[14] Malevsky, A. V., Spline-characteristic method for simulation of convective turbulence, J. Comput. Phys., 123, 466-466 (1996) · Zbl 0848.76064
[15] Bartello, P.; Thomas, S. J., The cost-effectiveness of semi-Lagrangian advection, Mon. Wea. Rev., 124, 2883-2883 (1996)
[16] Giraldo, F. X., The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids, J. Comput. Phys., 147, 114-114 (1998) · Zbl 0920.65070
[17] McGregor, J. L., Economical determination of departure points for semi-Lagrangian models, Mon. Wea. Rev., 121, 221-221 (1993)
[18] McDonald, A.; Bates, J. R., Improving the estimate of the departure point position in a two-time level semi-Lagrangian and semi-implicit scheme, Mon. Wea. Rev., 115, 737-737 (1987)
[19] McDonald, A., Accuracy of multi-upstream, semi-Lagrangian advective schemes, Mon. Wea. rev., 112, 1267-1267 (1984)
[20] Suli, E. S., Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53, 459-459 (1988) · Zbl 0637.76024
[21] Allievi, A.; Bermejo, R., Finite element modified method of characteristics for the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 32, 439-439 (2000) · Zbl 0955.76048
[22] Priestley, A., Exact projections and the Lagrange-Galerkin method: A realistic alternative to quadrature, J. Comput. Phys., 112, 316-316 (1994) · Zbl 0809.65097
[23] Achdou, Y.; Guermond, J. L., Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations, SIAM J. Numer. Anal., 37, 799-799 (2000) · Zbl 0966.76041
[24] Deville, M. O., Fischer, P. F., and Mund, E. H. (2002).High-Order Methods for Incompressible Fluid Flow. Cambridge University Press. · Zbl 1007.76001
[25] Momeni-Masuleh, S. H. (2001).Spectral Methods for the Three Field Formulation of Incompressible Fluid Flow. PhD thesis, Aberystwyth.
[26] Sherwin, S. J. (2003). A substepping Navier-Stokes splitting scheme for spectral/hp element discretisations. In Matuson, T., Ecer, A., Periaux, J., Satufka, N., and Fox, P. (eds.),Parallel Computational Fluid Dynamics: New Frontiers Multi-Disciplinary Applications, North-Holland, pp. 43-52. · Zbl 1074.76572
[27] Leriche, E.; Labrosse, G., High-order direct Stokes solvers with or without temporal splitting: numerical investigations of their comparative properties, SIAM J. Sci. Comput., 22, 4, 1386-1386 (2000) · Zbl 0972.35087
[28] Ghia, U.; Ghia, K. N.; Shin, C. T., High-Re solutions for the incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys., 48, 387-387 (1982) · Zbl 0511.76031
[29] Koseff, J. R.; Street, R. L., The lid-driven cavity flow: a synthesis of qualitative and quantitative observations, ASME J. Fluids Eng., 106, 390-390 (1984)
[30] Xu, J.; Xiu, D.; Karniadakis, G. E., A semi-Lagrangian method for turbulence simulations using mixed spectral discretizations, J. Sci. Comput., 17, 585-585 (2002) · Zbl 1028.76020
[31] Smolarkiewicz, P. K.; Pudykiewicz, J., A class of semi-Lagrangian approximations for fluids, J. Atmo. Sci., 49, 2082-2082 (1992) · Zbl 0774.62029
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