Skip to main content
SpringerLink
Log in

On the relationship between Semi-Lagrangian and Lagrange–Galerkin schemes

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

Abstract

Following a previous result stating their equivalence under constant advection speed, Semi-Lagrangian and Lagrange–Galerkin schemes are compared in this paper in the situation of variable coefficient advection equations. Once known that Semi-Lagrangian schemes can be proved to be equivalent to area-weighted Lagrange–Galerkin schemes via a suitable definition of the basis functions, we will further prove that area-weighted Lagrange–Galerkin schemes represent a “small” (more precisely, an \(O(\Delta t\))) perturbation of exact Lagrange–Galerkin schemes. This equivalence implies a general result of stability for Semi-Lagrangian schemes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Notes

  1. In fact, the method proposed in [14] applies this technique element! by element following the displacement of centroids; however, it is conceptually equivalent to apply the same technique to the whole basis function \(\phi _j\) provided it is piecewise polynomial and compactly supported as assumed in [14].

References

  1. Besse, N., Mehrenberger, M.: Convergence of classes of high-order semi-Lagrangian schemes for the Vlasov-Poisson system. Math. Comp. 77, 93–123 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cheng, C.Z., Knorr, G.: The integration of the Vlasov equation in configuration space. J. Comput. Phys. 22, 330–351 (1976)

    Article  Google Scholar 

  3. Courant, R., Isaacson, E., Rees, M.: On the solution of nonlinear hyperbolic differential equations by finite differences. Comm. Pure Appl. Math. 5, 243–255 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  4. 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. Num. Anal. 19, 871–885 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  5. Falcone, M., Ferretti, R.: Convergence analysis for a class of semi-Lagrangian advection schemes. SIAM J. Num. Anal. 35, 909–940 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Falcone, M., Ferretti, R.: Semi-Lagrangian schemes for Hamilton–Jacobi equations, discrete representation formulae and Godunov methods. J. Comput. Phys. 175, 559–575 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ferretti, R.: Equivalence of semi-Lagrangian and Lagrange–Galerkin schemes under constant advection speed. J. Comp. Math. 28, 461–473 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Giraldo, F.X.: Lagrange–Galerkin methods on spherical geodesic grids: the shallow water equations. J. Comput. Phys. 160, 336–368 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gonzáles Gutiérrez, L.M., Bermejo, R.: A semi-Lagrangian level set method for incompressible Navier–Stokes equations with free surface. Int. J. Numer. Meth. Fluids 49, 1111–1146 (2005)

    Article  Google Scholar 

  10. Houston, P., Süli, E.: Adaptive Lagrange–Galerkin methods for unsteady convection–diffusion problems. Math. Comp. 70, 77–106 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lin, S.-J., Rood, R.B.: Multi-dimensional flux-form semi-Lagrangian transport schemes. Mon. Wea. Rev. 124, 2046–2070 (1996)

    Article  Google Scholar 

  12. McDonald, A.: Accuracy of multiply-upstream semi-Lagrangian advective schemes. Mon. Wea. Rev. 112, 1267–1275 (1984)

    Article  Google Scholar 

  13. McDonald, A.: Accuracy of multiply-upstream semi-Lagrangian advective schemes II. Mon. Wea. Rev. 115, 1446–1450 (1987)

    Article  Google Scholar 

  14. Morton, K.W., Priestley, A., Süli, E.: Stability of the Lagrange–Galerkin method with non-exact, integration. Math. Model Numer. Anal. 22, 625–653 (1988)

    MATH  Google Scholar 

  15. Papoulis, A.: Signal Analysis. MacGraw-Hill, New York (1977)

    MATH  Google Scholar 

  16. Pironneau, O.: On the transport-diffusion algorithm and its application to the Navier–Stokes equations. Num. Math. 38, 309–332 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Priestley, A.: Exact projections and the Lagrange–Galerkin method: a realistic alternative to quadrature. J. Comput. Phys. 112, 316–333 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. Qiu, J.-M., Shu, C.-W.: Convergence of Godunov-type schemes for scalar conservation laws under large time steps. SIAM J. Num. Anal. 46, 2211–2237 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Russell, T.F., Celia, M.A.: An overview of research on Eulerian–Lagrangian localized adjoint methods (ELLAM). Adv. Water Res. 25, 1215–1231 (2002)

    Article  Google Scholar 

  20. Staniforth, A.N., Côtè, J.: Semi-Lagrangian integration schemes for atmospheric models—a review. Mon. Wea. Rev. 119, 2206–2223 (1991)

    Article  Google Scholar 

  21. Strain, J.: Semi-Lagrangian methods for level set equations. J. Comput. Phys. 151, 498–533 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  22. Süli, E.: Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier–Stokes equations. Num. Math. 53, 459–483 (1988)

    Article  MATH  Google Scholar 

  23. Wiin-Nielsen, A.: On the application of trajectory methods in numerical forecasting. Tellus 11, 180–196 (1959)

    Article  Google Scholar 

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

    Google Scholar 

Download references

Acknowledgments

I am indebted with at least two colleagues. The first is Rodolfo Bermejo who first suggested to me the idea of studying the equivalence of SL and LG schemes. The second is Corrado Falcolini for his kind help with Mathematica.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tre, L.go S. Leonardo Murialdo, 1, Rome, 00146, Italy

    Roberto Ferretti

Authors
  1. Roberto Ferretti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Ferretti.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ferretti, R. On the relationship between Semi-Lagrangian and Lagrange–Galerkin schemes. Numer. Math. 124, 31–56 (2013). https://doi.org/10.1007/s00211-012-0505-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-012-0505-5

Mathematics Subject Classification (2010)

Search

Navigation