A family of finite difference schemes for the convection-diffusion equation in two dimensions

  • E. Sousa
  • I. Sobey
Conference paper


The construction of finite difference schemes in two dimensions is more ambiguous than in one dimension. This ambiguity arises because different combinations of local nodal values are equally able to model local behaviour with the same order of accuracy. In this paper we outline an evolution operator for the two-dimensional convection-diffusion problem in an unbounded domain and use it as the source for obtaining a family of second order (Lax-Wendroff) schemes and third-order (Quickest) schemes not yet studied in the literature. Additionally we study and compare the stability of these second-order and third-order schemes using the von Neumann method.


Evolution Operator Finite Difference Scheme Unbounded Domain Quick Scheme Local Polynomial Interpolation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Beckers, J.-M. (1992): Analytical linear numerical stability conditions for an anisotropic three-dimensional advection-diffusion equation. SIAM J. Numer. Anal. 29, 701–713MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Davis, R.W., Moore, E.F. (1982): A numerical study of vortex shedding from rectangles. J. Fluid Mech. 116, 475–506MATHCrossRefGoogle Scholar
  3. [3]
    Gustafsson, B., Kreiss, H.-O., Oliger, J. (1995): Time dependent problems and difference methods. Wiley-Interscience, New YorkMATHGoogle Scholar
  4. [4]
    Hindmarsh, A.C., Gresho, P.M., Griffiths, D.F. (1984): The stability of explicit Euler time-integration for certain finite difference approximations of the multi-dimensional advection-diffusion equation. Internat. J. Numer. Methods Fluids 4, 853–897MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Kwok, Y.-K., Tam, K.-K. (1993): Stability analysis of three-level difference schemes for initial-boundary problems for multidimensional convective-diffusion equations. Comm. Numer. Methods Engrg. 9, 595–605MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Lax, P.D., Wendroff, B. (1964): Difference schemes for hyperbolic equations with high order of accuracy. Comm. Pure Appl. Math. 17, 381–398MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Leonard, B.P. (1979): A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput. Methods Appl. Mech. Engrg. 19, 59–98MATHCrossRefGoogle Scholar
  8. [8]
    Morton, K. W., Sobey, I.J. ( 1993): Discretization of a convection-diffusion equation. IMA J. Numer. Anal. 13, 141–160MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Richtmyer, R.D., Morton, K.W. (1967): Difference methods for initial-value problems. Wiley-Interscience, New YorkMATHGoogle Scholar
  10. [10]
    Siemieniuch, J., Gladwell, I. (1978): Analysis of explicit difference methods for the diffusion-convection equation. Internat. J. Numer. Methods Engrg. 12, 899–916MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Verwer, J.G., Sommeijer, B.P. (1997): Stability analysis of an odd-even-line hopscotch method for three-dimensional advection-diffusion problems. SIAM J. Numer. Anal. 34, 376–388MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Wesseling, P. (1996): von Neumann stability conditions for the convection-diffusion equation. IMA J. Numer. Anal. 16, 583–598MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • E. Sousa
    • 1
    • 2
  • I. Sobey
    • 3
  1. 1.Oxford University Computing LaboratoryOxfordUK
  2. 2.Departamento de MatemáticaUniversidade de CoimbraCoimbraPortugal
  3. 3.Computing LaboratoryOxford UniversityOxfordUK

Personalised recommendations