Advertisement

BIT Numerical Mathematics

, Volume 52, Issue 4, pp 981–1007 | Cite as

Fully discrete semi-Lagrangian methods for advection of differential forms

  • Holger Heumann
  • Ralf Hiptmair
  • Kun Li
  • Jinchao Xu
Article

Abstract

We study the discretization of linear transient transport problems for differential forms on bounded domains. The focus is on unconditionally stable semi-Lagrangian methods that employ finite element approximation on fixed meshes combined with tracking of the flow map. We derive these methods as finite element Galerkin approach to discrete material derivatives and discuss further approximations leading to fully discrete schemes.

We establish comprehensive a priori error estimates, in particular a new asymptotic estimate of order \(O(h^{r+1}\tau^{-\frac{1}{2}})\) for the L 2-error of semi-Lagrangian schemes with exact L 2-projection. Here, h is the spatial meshwidth, τ denotes the timestep, and r is the (full) polynomial degree of the piecewise polynomial discrete differential forms used as trial functions. Yet, numerical experiments hint that the estimates may still be sub-optimal for spatial discretization with lowest order discrete differential forms.

Keywords

Advection-diffusion problem Discrete differential forms Discrete Lie derivative Semi-Lagrangian methods 

Mathematics Subject Classification

65M60 65M25 

References

  1. 1.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (New Ser.) 47(2), 281–354 (2010) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Baker, M.D., Süli, E., Ware, A.F.: Stability and convergence of the spectral Lagrange-Galerkin method for mixed periodic/non-periodic convection-dominated diffusion problems. IMA J. Numer. Anal. 19(4), 637–663 (1999) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Benson, D.J.: Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Methods Appl. Mech. Eng. 99(2–3), 235–394 (1992) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bercovier, M., Pironneau, O., Sastri, V.: Finite elements and characteristics for some parabolic-hyperbolic problems. Appl. Math. Model. 7(2), 89–96 (1983) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bossavit, A.: In: Applied Differential Geometry (2005). http://butler.cc.tut.fi/~bossavit/BackupICM/Compendium.html Google Scholar
  7. 7.
    Bossavit, A.: Discretization of electromagnetic problems: the “generalized finite differences”. In: Schilders, W.H.A., ter Maten, W.J.W. (eds.) Numerical Methods in Electromagnetics. Handbook of Numerical Analysis, vol. XIII, pp. 443–522. Elsevier, Amsterdam (2005) Google Scholar
  8. 8.
    Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4. North-Holland, Amsterdam (1978) MATHCrossRefGoogle Scholar
  9. 9.
    Dawson, J.M.: Particle simulation of plasmas. Rev. Mod. Phys. 55(2), 403–447 (1983) CrossRefGoogle Scholar
  10. 10.
    Douglas, J. Jr., 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. Numer. Anal. 19(5), 871–885 (1982) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Ewing, R.E., Wang, H.: A summary of numerical methods for time-dependent advection-dominated partial differential equations. J. Comput. Appl. Math. 128(1–2), 423–445 (2001) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Ewing, R.E., Russell, T.F., Wheeler, M.F.: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Methods Appl. Mech. Eng. 47(1–2), 73–92 (1984) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Frankel, T.: The Geometry of Physics: An Introduction. Cambridge University Press, Cambridge (1998) Google Scholar
  14. 14.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, San Diego (1981) MATHGoogle Scholar
  15. 15.
    Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964) MATHGoogle Scholar
  16. 16.
    Hasbani, Y., Livne, E., Bercovier, M.: Finite elements and characteristics applied to advection-diffusion equations. Comput. Fluids 11(2), 71–83 (1983) MATHCrossRefGoogle Scholar
  17. 17.
    Heumann, H., Hiptmair, R.: Convergence of lowest order semi-Lagrangian schemes. Report 2011-47, SAM, ETH Zürich (2011). Submitted to Found. Comp. Math. Google Scholar
  18. 18.
    Heumann, H., Hiptmair, R.: Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete Contin. Dyn. Syst. 29(4), 1471–1495 (2011) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Heumann, H., Hiptmair, R.: Refined convergence theory for semi-Lagrangian schemes for pure advection. Technical report 2011-60, Seminar for Applied Mathematics, ETH Zürich (2011). http://www.sam.math.ethz.ch/reports/2011/60
  20. 20.
    Heumann, H., Hiptmair, R., Xu, J.: A semi-Lagrangian method for convection of differential forms. Technical report 2009-09, Seminar for Applied Mathematics, ETH Zürich (2009). http://www.sam.math.ethz.ch/reports/2009/09
  21. 21.
    Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Houston, P., Schwab, C., Süli, E.: Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39(6), 2133–2163 (2002) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Hughes, T.J.R., Brooks, A.: A multidimensional upwind scheme with no crosswind diffusion. In: Finite Element Methods for Convection Dominated Flows. AMD, vol. 34, pp. 19–35. Am. Soc. Mech. Engrs., New York (1979) Google Scholar
  24. 24.
    Jack, R.O.: Convergence properties of Lagrangian-Galerkin method with and without exact integration. Technical report OUCL report 87/10, Oxford (1987) Google Scholar
  25. 25.
    Johnson, C.: A new approach to algorithms for convection problems which are based on exact transport + projection. Comput. Methods Appl. Mech. Eng. 100(1), 45–62 (1992) MATHCrossRefGoogle Scholar
  26. 26.
    Lasaint, P., Raviart, P.-A.: On a finite element method for solving the neutron transport equation. In: In Proc. Sympos., Math. Res. Center, Univ. of Wisconsin-Madison, vol. 33, pp. 89–123. Academic Press, New York (1974) Google Scholar
  27. 27.
    Morton, K.W., Priestley, A., Süli, E.: Stability of the Lagrange-Galerkin method with nonexact integration. RAIRO Modél. Math. Anal. Numér. 22(4), 625–653 (1988) MATHGoogle Scholar
  28. 28.
    Morton, K.W., Süli, E.: Evolution-Galerkin methods and their supraconvergence. Numer. Math. 71(3), 331–355 (1995) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Nédélec, J.-C.: Mixed finite elements in R 3. Numer. Math. 35(3), 315–341 (1980) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Nédélec, J.-C.: A new family of mixed finite elements in R 3. Numer. Math. 50(1), 57–81 (1986) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Pironneau, O.: On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38(3), 309–332 (1981/1982) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Priestley, A.: Exact projections and the Lagrange-Galerkin method: a realistic alternative to quadrature. J. Comput. Phys. 112(2), 316–333 (1994) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Tech. rep. LA-UR-73-479, Los Alamos National Laboratory, Los Alamos, NM (1973) Google Scholar
  34. 34.
    Rieben, R.N., White, D.A., Wallin, B.K., Solberg, J.M.: An arbitrary Lagrangian-Eulerian discretization of MHD on 3D unstructured grids. J. Comput. Phys. 226(1), 534–570 (2007) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Scheja, G., Storch, U.: Lehrbuch der Algebra. Teil 2. Mathematische Leitfäden. Teubner, Stuttgart (1988) Google Scholar
  36. 36.
    Schwarz, G.: Hodge Decomposition—A Method for Solving Boundary Value Problems. Lecture Notes in Mathematics, vol. 1607. Springer, Berlin (1995) MATHGoogle Scholar
  37. 37.
    Staniforth, A., Côté, J.: Semi-Lagrangian integration schemes for atmospheric models: a review. Mon. Weather Rev. 119, 2206–2223 (1991) CrossRefGoogle Scholar
  38. 38.
    Süli, E., Ware, A.: A spectral method of characteristics for hyperbolic problems. SIAM J. Numer. Anal. 28(2), 423–445 (1991) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2012

Authors and Affiliations

  • Holger Heumann
    • 1
  • Ralf Hiptmair
    • 1
  • Kun Li
    • 2
  • Jinchao Xu
    • 3
  1. 1.Seminar for Applied MathematicsSwiss Federal Institute of TechnologyZurichSwitzerland
  2. 2.School of Mathematical SciencesPeking UniversityBeijingChina
  3. 3.Department of MathematicsPenn State UniversityUniversity ParkUSA

Personalised recommendations