Top

Journal of Scientific Computing

Published in:

28-03-2015

High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations

Authors: Elena Celledoni, Bawfeh Kingsley Kometa, Olivier Verdier

Published in: Journal of Scientific Computing | Issue 1/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We propose a class of semi-Lagrangian methods of high approximation order in space and time, based on spectral element space discretizations and exponential integrators of Runge–Kutta type. The methods were presented in Celledoni and Kometa (J Sci Comput 41(1):139–164, 2009) for simpler convection–diffusion equations. We discuss the extension of these methods to the Navier–Stokes equations, and their implementation using projections. Semi-Lagrangian methods up to order three are implemented and tested on various examples. The good performance of the methods for convection-dominated problems is demonstrated with numerical experiments.

previous article A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains

next article Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

We also assume that the characteristic paths are integrated to high accuracy.

Extension to higher order methods is straightforward (See [30] and references therein).

See 6.2 for the definitions of these norms.

The projection does not need to be orthogonal, but should be guaranteed not to compromise the order of the method, an orthogonal projection will have this property. The target of the orthogonal projection map on the discrete divergence-free subspace is the element of shortest distance to the point which is projected. Since $y_{n+1}=\Pi Y_s$ and $\Vert y(t_{n+1})-Y_s\Vert = {\mathcal {O}}(h^{r+1}),$ where $r$ is the order of the integration method, then

$$\begin{aligned} \Vert y(t_{n+1})-y_{n+1}\Vert \le \Vert y(t_{n+1})-Y_s\Vert +\Vert Y_s-y_{n+1}\Vert = {\mathcal {O}}(h^{r+1}), \end{aligned}$$

because

$$\begin{aligned} \Vert y_{n+1}-Y_s\Vert \le \Vert y(t_{n+1})-Y_s\Vert . \end{aligned}$$

Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)MathSciNetCrossRefMATH

Ascher, U.M., Ruuth, S.J., Wetton, B.T.R.: Implicit–explicit methods for time-dependent partial differential equations. SIAM J. Numer. Anal. 32(3), 797–823 (1995)MathSciNetCrossRefMATH

Brown, D.L., Minion, M.L.: Performance of under-resolved two-dimensional incompressible flow simulations. J. Comput. Phys. 122(1), 165–183 (1995)MathSciNetCrossRefMATH

Bruneau, C.-H., Saad, M.: The 2d lid-driven cavity problem revisited. Comput. Fluids 35(3), 326–348 (2006)CrossRefMATH

Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer Series in Computational Physics. Springer, New York (1988)CrossRef

Celledoni, E.: Eulerian and semi-Lagrangian commutator-free exponential integrators. CRM Proc. Lect. Notes 39, 77–90 (2005)MathSciNet

Celledoni, E., Kometa, B.K.: Semi-Lagrangian Runge–Kutta exponential integrators for convection dominated problems. J. Sci. Comput. 41(1), 139–164 (2009)MathSciNetCrossRefMATH

Celledoni, E., Kometa, B.K.: Order conditions for semi-Lagrangian Runge-Kutta exponential integrators, Preprint series: Numerics, 04/2009, Department of Mathematics, NTNU, Trondheim, Norway. http://www.math.ntnu.no/preprint/numerics/N4-2009.pdf (2009)

Celledoni, E., Kometa, B.K.: Semi-Lagrangian multistep exponential integrators for index 2 differential-algebraic systems. J. Comput. Phys. 230(9), 3413–3429 (2011)MathSciNetCrossRefMATH

10.

Childs, P.N., Morton, K.W.: Characteristic Galerkin methods for scalar conservation laws in one dimension. SIAM J. Numer. Anal. 27, 553–594 (1990)MathSciNetCrossRefMATH

11.

Chorin, A.J.: Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762 (1968)MathSciNetCrossRefMATH

12.

Chorin, A.J.: On the convergence of discrete approximations to the Navier–Stokes equations. Math. Comput. 23, 341–353 (1969)MathSciNetCrossRefMATH

13.

Deville, M.O., Fischer, P.F., Mund, E.H.: High-Order Methods for Incompressible Fluid Flow. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002)CrossRef

14.

Eich-Soellner, E., Führer, C.: Numerical Methods in Multibody Dynamics, European Consortium for Mathematics in Industry. B. G. Teubner, Stuttgart (1998)CrossRef

15.

Falcone, M., Ferretti, R.: Convergence analysis for a class of high-order semi-Lagrangian advection schemes. SIAM J. Numer. Anal. 35(3), 909–940 (1998). (electronic)MathSciNetCrossRefMATH

16.

Falcone, M., Ferretti, R.: Semi-Lagrangian approximation schemes for linear and Hamilton-Jacobi equations. SIAM (to appear)

17.

Fischer, P., Mullen, J.: Filter-based stabilization of spectral element methods. C. R. Acad. Sci. Paris Sér. I Math. 332(3), 265–270 (2001)MathSciNetCrossRefMATH

18.

Fischer, P.F.: An overlapping Schwarz method for spectral element solution of the incompressible Navier–Stokes equations. J. Comput. Phys. 133(1), 84–101 (1997)MathSciNetCrossRefMATH

19.

Fischer, P.F., Kruse, G.W., Loth, F.: Spectral element method for transitional flows in complex geometries. J. Sci. Comput. 17(1–4), 81–98 (2002)MathSciNetCrossRefMATH

20.

Ghia, U., Ghia, K.N., Shin, C.T.: High re-solutions for incompressible flow using the Navier–stokes equations and a multigrid method. J. Comput. Phys. 48(3), 387–411 (1982)CrossRefMATH

21.

Giraldo, F.X.: The Lagrange–Galerkin spectral element method on unstructured quadrilateral grids. J. Comput. Phys. 147(1), 114–146 (1998)MathSciNetCrossRefMATH

22.

Giraldo, F.X., Perot, J.B., Fischer, P.F.: A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations. J. Comput. Phys. 190(2), 623–650 (2003)MathSciNetCrossRefMATH

23.

Guermond, J.L., Shen, J.: A new class of truly consistent splitting schemes for incompressible flows. J. Comput. Phys. 192(1), 262–276 (2003)MathSciNetCrossRefMATH

24.

Guermond, J.L., Minev, P.D., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006). doi:10.1016/j.cma.2005.10.010 MathSciNetCrossRefMATH

25.

Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration, Second ed., Springer Series in Computational Mathematics, vol. 31, Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin (2006)

26.

Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II, Second ed., Springer Series in Computational Mathematics. Springer, Berlin (1996)CrossRef

27.

Kanevsky, A., Carpenter, M.H., Gottlieb, D., Hesthaven, J.S.: Application of implicit–explicit high order Runge–Kutta methods to discontinuous-Galerkin schemes. J. Comput. Phys. 225(2), 1753–1781 (2007)MathSciNetCrossRefMATH

28.

Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003)MathSciNetCrossRefMATH

29.

Kometa, B.K.: Semi-Lagrangian methods and new integration schemes for convection-dominated problems, PhD thesis at NTNU, ISSN 1503–8181; 2011:270, Norwegian University of Science and Technology, Department of Mathematical Sciences, http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-15729 (2011)

30.

Kvarving, A.M.: Splitting schemes for the unsteady Stokes equations: a comparison study, Preprint series: Numerics, 05/2010, Department of Mathematics, NTNU, Trondheim, Norway, http://www.math.ntnu.no/preprint/numerics/N5-2010.pdf (2010)

31.

Maday, Y., Patera, A.T., Rønquist, E.M.: An operator-integration-factor splitting method for time-dependent problems: application to incompressible fluid flow. J. Sci. Comput. 5(4), 263–292 (1990)MathSciNetCrossRefMATH

32.

Morton, K.W.: Generalised Galerkin methods for hyperbolic problems. Computer Methods Appl Mech Eng 52, 847–871 (1985)MathSciNetCrossRefMATH

33.

Rannacher, R.: On Chorin’s projection method for the incompressible Navier–Stokes equations, Lecture Notes in Mathematics, vol. 1530. Springer, Berlin (1991)

34.

Restelli, M., Bonaventura, L., Sacco, R.: A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows. J. Comput. Phys. 216(1), 195–215 (2006)MathSciNetCrossRefMATH

35.

Shen, J.: On error estimates of projection methods for Navier–Stokes equations: first-order schemes. SIAM J. Numer. Anal. 29(1), 57–77 (1992)MathSciNetCrossRefMATH

36.

Témam, R.: Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionnaires. II. Arch. Ration. Mech. Anal. 33, 377–385 (1969)CrossRefMATH

37.

Xiu, D., Karniadakis, G.E.: A semi-Lagrangian high-order method for Navier–Stokes equations. J. Comput. Phys. 172(2), 658–684 (2001)MathSciNetCrossRefMATH

Title: High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations
Authors: Elena Celledoni
Bawfeh Kingsley Kometa
Olivier Verdier
Publication date: 28-03-2015
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2016
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-015-0015-6

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2016

Petviashvilli’s Method for the Dirichlet Problem

Fast Numerical Contour Integral Method for Fractional Diffusion Equations

Adaptive Gradient-Augmented Level Set Method with Multiresolution Error Estimation

On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow

Towards Optimized Schwarz Methods for the Navier–Stokes Equations

A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings

Premium Partner