Advertisement

Journal of Scientific Computing

, Volume 57, Issue 1, pp 213–228 | Cite as

Stability of Some Generalized Godunov Schemes With Linear High-Order Reconstructions

  • Roberto Ferretti
Article

Abstract

Classical Semi-Lagrangian schemes have the advantage of allowing large time steps, but fail in general to be conservative. Trying to keep the advantages of both large time steps and conservative structure, Flux-Form Semi-Lagrangian schemes have been proposed for various problems, in a form which represent (at least in a single space dimension) a high-order, large time-step generalization of the Godunov scheme. At the theoretical level, a recent result has shown the equivalence of the two versions of Semi-Lagrangian schemes for constant coefficient advection equations, while, on the other hand, classical Semi-Lagrangian schemes have been proved to be strictly related to area-weighted Lagrange–Galerkin schemes for both constant and variable coefficient equations. We address in this paper a further issue in this theoretical framework, i.e., the relationship between stability of classical and of Flux-Form Semi-Lagrangian schemes. We show that the stability of the former class implies the stability of the latter, at least in the case of the one-dimensional linear continuity equation.

Keywords

Godunov schemes Semi-Lagrangian schemes Flux-Form Semi-Lagrangian schemes Stability 

Mathematics Subject Classification

65M12 65M08 65M25 

References

  1. 1.
    Crouseilles, N., Mehrenberger, M., Sonnendrücker, E.: Conservative Semi-Lagrangian schemes for Vlasov equations. J. Comp. Phys. 229, 1927–1953 (2010)MATHCrossRefGoogle Scholar
  2. 2.
    Falcone, M., Ferretti, R.: Semi-Lagrangian schemes for linear and Hamilton–Jacobi equations, SIAM (to appear)Google Scholar
  3. 3.
    Ferretti, R.: Equivalence of Semi-Lagrangian and Lagrange–Galerkin schemes under constant advection speed. J. Comp. Math. 28, 461–473 (2010)MathSciNetMATHGoogle Scholar
  4. 4.
    Ferretti, R.: On the relationship between semi-Lagrangian and Lagrange-Galerkin schemes. Num. Math. (2012). doi: 10.1007/s00211-012-0505-5
  5. 5.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser Verlag, Basel (1990)MATHGoogle Scholar
  6. 6.
    Lin, S.-J., Rood, R.B.: Multi-dimensional Flux-Form Semi-Lagrangian transport schemes. Mon. Wea. Rev. 124, 2046–2070 (1996)CrossRefGoogle Scholar
  7. 7.
    Morton, K.W.: On the analysis of finite volume methods for evolutionary problems. SIAM J. Num. Anal. 35, 2195–2222 (1998)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    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)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Qiu, J.-M., Shu, C.-W.: Conservative high order Semi-Lagrangian finite difference WENO methods for advection in incompressible flow. J. Comput. Phys. 230, 863–889 (2011)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Restelli, M., Bonaventura, L., Sacco, R.: A Semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows. J. Comput. Phys. 216, 195–215 (2006)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di Roma TreRomaItaly

Personalised recommendations