Flux-Splitting Schemes for Parabolic Problems

  • Petr N. Vabishchevich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


To solve numerically boundary value problems for parabolic equations with mixed derivatives, the construction of difference schemes with prescribed quality faces essential difficulties. In parabolic problems, some possibilities are associated with the transition to a new formulation of the problem, where the fluxes (derivatives with respect to a spatial direction) are treated as unknown quantities. In this case, the original problem is rewritten in the form of a boundary value problem for the system of equations in the fluxes. This work deals with studying schemes with weights for parabolic equations written in the flux coordinates. Unconditionally stable flux locally one-dimensional schemes of the first and second order of approximation in time are constructed for parabolic equations without mixed derivatives. A peculiarity of the system of equations written in flux variables for equations with mixed derivatives is that there do exist coupled terms with time derivatives.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Marchuk, G.I.: Handbook of Numerical Analysis, Splitting and alternating direction methods, vol. I. Elsevier Science Publishers B.V., North-Holland (1990)Google Scholar
  2. 2.
    Samarskii, A.A., Vabishchevich, P.N.: Additive schemes for problems of mathematical physics. Nauka (1999) (in Russian)Google Scholar
  3. 3.
    Degtyarev, L.M., Favorskii, A.P.: A flow variant of the sweep method. USSR Comput. Math. Math. Phys. 8(3), 252–261 (1968)CrossRefGoogle Scholar
  4. 4.
    Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Heidelberg (1991)CrossRefzbMATHGoogle Scholar
  5. 5.
    Roberts, J.E., Thomas, J.M.: Handbook of Numerical Analysis, Mixed and hybrid methods. Amsterdam, vol. II. Elsevier Science Publishers B.V., North-Holland (1991)Google Scholar
  6. 6.
    Samarskii, A.A.: The theory of difference schemes. Marcel Dekker Inc., New York (2001)CrossRefzbMATHGoogle Scholar
  7. 7.
    Matus, P., Rybak, I.: Difference schemes for elliptic equations with mixed derivatives. Computational Methods in Applied Mathematics 4(4), 494–505 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Samarskii, A.A., Matus, P.P., Vabishchevich, P.N.: Difference schemes with operator factors. Kluwer Academic Pub. (2002)Google Scholar
  9. 9.
    McKee, S., Mitchell, A.R.: Alternating direction methods for parabolic equations in two space dimensions with a mixed derivative. The Computer Journal 13(1), 81–86 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hout, K.J., Mishra, C.: Stability of the modified craig-sneyd scheme for twodimensional convection-diffusion equations with mixed derivative term. Mathematics and Computers in Simulation 81(11), 2540–2548 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Petr N. Vabishchevich
    • 1
    • 2
  1. 1.Nuclear Safety InstituteMoscowRussia
  2. 2.North-Eastern Federal UniversityYakutskRussia

Personalised recommendations