×

A semi-Lagrangian high-order method for Navier-Stokes equations. (English) Zbl 1028.76026

Summary: We present a semi-Lagrangian method for advection-diffusion and incompressible Navier-Stokes equations. The focus is on constructing stable schemes of second-order temporal accuracy, as this is a crucial element for the successful application of semi-Lagrangian methods to turbulence simulations. We implement the method in the context of unstructured spectral/\(hp\) element discretization, which allows for efficient search-interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal (advection) error of the form \({\mathcal O}(\Delta t^k+{ \Delta x^{p+1} \over\Delta t})\). We present numerical results that validate this error estimate for the advection-diffusion equation, and we document that such estimate is also valid for the Navier-Stokes equations at moderate or high Reynolds number. Two- and three-dimensional laminar and transitional flow simulations suggest that semi-Lagrangian schemes are more efficient than their Eulerian counterparts for high-order discretizations on nonuniform grids.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76M22 Spectral methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] 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 (2000) · Zbl 0966.76041
[2] Allievi, A.; Bermejo, R., Finite element modified method of characteristics for the Navier-Stokes equations, Int. J. Numer. Methods Fluids, 32, 439 (2000) · Zbl 0955.76048
[3] Barkley, D.; Henderson, R. D., Three-dimensional Floquet stability analysis of the wake of a circular cylinder, J. Fluid Mech., 322, 215 (1996) · Zbl 0882.76028
[4] Bartello, P.; Thomas, S. J., The cost-effectiveness of semi-Lagrangian advection, Mon. Wea. Rev., 124, 2883 (1996)
[5] Brassington, G.; Sanderson, B., Semi-Lagrangian and COSMIC advection in flows with rotation or deformation, Atmos.-Ocean., 37, 369 (1999)
[6] Chu, D.; Karniadakis, G. E., A direct numerical simulation of laminar and turbulent flow over riblet-mounted surfaces, J. Fluid Mech., 250, 1 (1993)
[7] Darekar, R. M.; Sherwin, S. J., Flow past a square section cylinder with a wavy stagnation face, J. Fluid Mech., 426, 263 (2001) · Zbl 1016.76015
[8] Davis, R. W.; Moore, E. F.; Purtell, L. P., A numerical-experimental study of confined flow around rectangular cylinders, Phys. Fluids, 27, 46 (1984)
[9] Falcone, M.; Ferretti, R., Convergence analysis for a class of high-order semi-Lagrangian advection schemes, SIAM J. Numer. Anal., 35, 909 (1998) · Zbl 0914.65097
[10] Franke, R.; Rodi, W.; Schönung, B., Numerical calculation of laminar vortex-shedding flow past cylinders, J. Wind Eng. Indust. Aero., 35, 237 (1990)
[11] Gear, G. W., Numerical Initial Value Problems in Ordinary Differential Equations (1971)
[12] 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 (1982) · Zbl 0511.76031
[13] Giraldo, F. X., The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids, J. Comput. Phys., 147, 114 (1998) · Zbl 0920.65070
[14] Huang, L. C., Problem of the pressure correction projection method-additional notes to our 6ISCFD paper, in Institute of Computational Mathematics and Scientific/Engineering Computing (1995), Chinese Academy of Sciences
[15] 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 (1984) · Zbl 0547.76038
[16] Karniadakis, G. E.; Israeli, M.; Orszag, S. A., High-order splitting methods for the incompressible Navier-Stokes equations, J. Comput. Phys., 97, 414 (1991) · Zbl 0738.76050
[17] Karniadakis, G. E.; Orszag, S. A., Nodes, modes, and flow codes, Physics Today, 34 (1993)
[18] Karniadakis, G. E.; Sherwin, S. J., Spectral/hp Element Methods for CFD (1999) · Zbl 0954.76001
[19] Koseff, J. R.; Street, R. L., The lid-driven cavity flow: a synthesis of qualitative and quantitative observations, ASME J. Fluids Eng., 106, 390 (1984)
[20] Malevsky, A. V., Spline-characteristic method for simulation of convective turbulence, J. Comput. Phys., 123, 466 (1996) · Zbl 0848.76064
[21] Malevsky, A. V.; Thomas, S. J., Parallel algorithms for semi-Lagrangian advection, Int. J. Numer. Meth. Fluids, 25, 455 (1997) · Zbl 0910.76063
[22] Xia, Ma; Karamanos, G. S.; Karniadakis, G. E., Dynamics and low-dimensionality of turbulent near-wake, J. Fluid Mech., 410, 29 (2000) · Zbl 0987.76041
[23] McDonald, A., Accuracy of multi-upstream, semi-Lagrangian advective schemes, Mon. Wea. Rev., 112, 1267 (1984)
[24] 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 (1987)
[25] McGregor, J. L., Economical determination of departure points for semi-Lagrangian models, Mon. Wea. Rev., 121, 221 (1993)
[26] Okajima, A., Strouhal numbers of rectangular cylinders, J. Fluid Mech., 123, 379 (1982)
[27] Oliveira, A.; Baptista, A. M., A comparison of integration and interpolation Eulerian-Lagrangian methods, Int. J. Numer. Methods Fluids, 21, 183 (1995) · Zbl 0841.76041
[28] Pironneau, O., On the transport-diffusion algorithm and its applications to the Navier-Stokes equations, Numer. Math., 38, 309 (1982) · Zbl 0505.76100
[29] Priestley, A., Exact projections and the Lagrange-Galerkin method: A realistic alternative to Quadrature, J. Comput. Phys., 112, 316 (1994) · Zbl 0809.65097
[30] Pudykiewicz, J.; Benoit, R.; Staniforth, A., Preliminary results from a partial LRTAP model based on an existing meteorological forecast model, Atmos.-Ocean, 23, 267 (1985)
[31] Robert, A., A stable numerical integration scheme for the primitive meteorological equations, Atmos.-Ocean., 19, 35 (1981)
[32] Saha, A. K.; Muralidhar, K.; Biswas, G., Transition and chaos in two-dimensional flow past a square cylinder, J. Eng. Mech., 126, 523 (2000) · Zbl 0980.76060
[33] Shen, J., Hopf bifurcation of the unsteady regularized driven cavity flow, J. Comput. Phys., 95, 228 (1991) · Zbl 0725.76059
[34] Smolarkiewicz, P. K.; Pudykiewicz, J., A class of semi-Lagrangian approximations for fluids, J. Atms. Sci., 49, 2082 (1992)
[35] Sohankar, A.; Norberg, C.; Davidson, L., Low-Reynolds-number flow around a square cylinder at incidence: study of blockage, onset of vortex shedding and outlet boundary condition, Int. J. Numer. Methods Fluids, 26, 39 (1998) · Zbl 0910.76067
[36] Sohankar, A.; Norberg, C.; Davidson, L., Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers, Phys. Fluids, 11, 288 (1999) · Zbl 1147.76502
[37] Staniforth, A.; Côté, J., Semi-Lagrangian integration schemes for atmospheric models-A review, Mon. Wea. Rev., 119, 2206 (1991)
[38] Süli, E., Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations, Numer. Math., 53, 459 (1988) · Zbl 0637.76024
[39] Williamson, C. H.K.; Roshko, A., Vortex formation in the wake of an oscillating cylinder, J. Fluid Struct., 2, 355 (1988)
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.