Advertisement

Domain Decomposition Scheme for First-Order Evolution Equations with Nonselfadjoint Operators

  • Petr VabishchevichEmail author
  • Petr Zakharov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 45)

Abstract

Domain decomposition iterative methods and implicit schemes are usually used for solving evolution equations. An alternative approach is based on constructing non-iterative method based on special schemes of splitting into subdomains. Such regional-additive schemes are based on the general theory of additive operator-difference schemes. Domain decomposition analogues of the classical schemes of alternating direction method, locally one-dimensional schemes, factored schemes, and regularized vector-additive schemes are used here. The main results in the literature are obtained for time-dependent problems with selfadjoint second-order elliptic operators. This paper discusses the Cauchy problem for first-order evolution equations with nonnegative nonselfadjoint operators in a finite-dimensional Hilbert space. Based on the partition of unity, we have constructed nonnegativity preserving decomposition operators for the respective operator term in the equation. We construct unconditionally stable additive domain decomposition schemes based on the principle of regularization of operator-difference schemes and vector-additive schemes.

Keywords

First-order evolution equations Parabolic partial differencial equation Domain decomposition method Difference scheme. 

Notes

Acknowledgements

This research was supported by the NEFU Development Program for 2010–2019.

References

  1. 1.
    Abrashin, V.: A variant of the method of variable directions for the solution of multi-dimensional problems of mathematical-physics. I. Differ. Equat. 26(2), 243–250 (1990)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abrashin, V., Vabishchevich, P.: Vector additive schemes for second-order evolution equations. Differ. Equat. 34(12), 1673–1681 (1998)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Cai, X.C.: Additive Schwarz algorithms for parabolic convection-diffusion equations. Numer. Math. 60(1), 41–61 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Cai, X.C.: Multiplicative Schwarz methods for parabolic problems. SIAM J. Sci. Comput. 15(3), 587–603 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dryja, M.: Substructuring methods for parabolic problems. In: Glowinski, R., Kuznetsov, Y.A., Meurant, G.A., Périaux, J., Widlund, O. (eds.) Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations. SIAM, Philadelphia, PA (1991)Google Scholar
  6. 6.
    Kuznetsov, Y.: New algorithms for approximate realization of implicit difference schemes. Sov. J. Numer. Anal. Math. Model. 3(2), 99–114 (1988)zbMATHGoogle Scholar
  7. 7.
    Kuznetsov, Y.: Overlapping domain decomposition methods for FE-problems with elliptic singular perturbed operators. Fourth international symposium on domain decomposition methods for partial differential equations, Proc. Symp., Moscow/Russ. 1990, 223–241 (1991) (1991)Google Scholar
  8. 8.
    Laevsky, Y.: Domain decomposition methods for the solution of two-dimensional parabolic equations. In: Variational-difference methods in problems of numerical analysis, vol. 2, pp. 112–128. Comp. Cent. Sib. Branch, USSR Acad. Sci., Novosibirsk (1987). In RussianGoogle Scholar
  9. 9.
    Lax, P.D.: Linear algebra and its applications. 2nd edn. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs and Tracts, xvi, 376 p. Wiley, New York (2007)Google Scholar
  10. 10.
    Marchuk, G.: Splitting and alternating direction methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. I, pp. 197–462. North-Holland, Amsterdam (1990)Google Scholar
  11. 11.
    Mathew, T.: Domain decomposition methods for the numerical solution of partial differential equations. Lecture Notes in Computational Science and Engineering, vol. 61, xiii, 764 p. Springer, Berlin (2008)Google Scholar
  12. 12.
    Quarteroni, A., Valli, A.: Domain decomposition methods for partial differential equations. Numerical Mathematics and Scientific Computation, xv, 360 p. Clarendon Press, Oxford (1999)Google Scholar
  13. 13.
    Samarskii, A.: The theory of difference schemes. Pure and Applied Mathematics, Marcel Dekker, vol. 240, 786 p. Marcel Dekker, New York (2001)Google Scholar
  14. 14.
    Samarskii, A., Matus, P., Vabishchevich, P.: Difference schemes with operator factors. Mathematics and Its Applications (Dordrecht), vol. 546, x, 384 p. Kluwer Academic, Dordrecht (2002)Google Scholar
  15. 15.
    Samarskii, A., Nikolaev, E.: Numerical Methods for Grid Equations. Birkhäuser, Basel (1989)zbMATHCrossRefGoogle Scholar
  16. 16.
    Samarskii, A., Vabishchevich, P.: Vector additive schemes of domain decomposition for parabolic problems. Differ. Equat. 31(9), 1522–1528 (1995)MathSciNetGoogle Scholar
  17. 17.
    Samarskii, A., Vabishchevich, P.: Factorized finite-difference schemes for the domain decomposition in convection-diffusion problems. Differ. Equat. 33(7), 972–979 (1997)MathSciNetGoogle Scholar
  18. 18.
    Samarskii, A., Vabishchevich, P.: Regularized additive full approximation schemes. Doklady. Math. 57(1), 83–86 (1998)Google Scholar
  19. 19.
    Samarskii, A., Vabishchevich, P.: Additive Schemes for Problems of Mathematical Physics (Additivnye skhemy dlya zadach matematicheskoj fiziki), 320 p. Nauka, Moscow (1999). In RussianzbMATHGoogle Scholar
  20. 20.
    Samarskii, A., Vabishchevich, P.: Domain decomposition methods for parabolic problems. In: Lai, C.H., Bjorstad, P., Gross, M., Widlund, O. (eds.) Eleventh International Conference on Domain Decomposition Methods, pp. 341–347. DDM.org (1999)Google Scholar
  21. 21.
    Samarskii, A., Vabishchevich, P.: Numerical Methods for Solution of Convection-Diffusion Problems (Chislennye metody resheniya zadach konvekcii-diffuzii), 247 p. URSS, Moscow (1999). In RussianGoogle Scholar
  22. 22.
    Samarskii, A., Vabishchevich, P., Matus, P.: Stability of vector additive schemes. Doklady Math. 58(1), 133–135 (1998)MathSciNetGoogle Scholar
  23. 23.
    Samarskii, A.A., Vabishchevich, P.N.: Regularized difference schemes for evolutionary second order equations. Math. Model Methods Appl. Sci. 2(3), 295–315 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Smith, B.: Domain decomposition. Parallel Multilevel Methods for Elliptic Partial Differential Equations, xii, 224 p. Cambridge University Press, Cambridge (1996)Google Scholar
  25. 25.
    Toselli, A., Widlund, O.: Domain decomposition methods – algorithms and theory. Springer Series in Computational Mathematics, vol 34, xv, 450 p. Springer, Berlin (2005)Google Scholar
  26. 26.
    Vabishchevich, P.: Difference schemes with domain decomposition for solving nonstationary problems. U.S.S.R. Comput. Math. Math. Phys. 29(6), 155–160 (1989)Google Scholar
  27. 27.
    Vabishchevich, P.: Regional-additive difference schemes for nonstationary problems of mathematical physics. Mosc. Univ. Comput. Math. Cybern. (3), 69–72 (1989)MathSciNetGoogle Scholar
  28. 28.
    Vabishchevich, P.: Parallel domain decomposition algorithms for time-dependent problems of mathematical physics. Advances in Numerical Methods and Applications, pp. 293–299. World Schientific, Singapore (1994)Google Scholar
  29. 29.
    Vabishchevich, P.: Regionally additive difference schemes with a stabilizing correction for parabolic problems. Comput. Math. Math. Phys. 34(12), 1573–1581 (1994)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Vabishchevich, P.: Finite-difference domain decomposition schemes for nonstationary convection-diffusion problems. Differ. Equat. 32(7), 929–933 (1996)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Vabishchevich, P.: Vector additive difference schemes for first-order evolutionary equations. Comput. Math. Math. Phys. 36(3), 317–322 (1996)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Vabishchevich, P.: Domain decomposition methods with overlapping subdomains for the time-dependent problems of mathematical physics. Comput. Methods Appl. Math. 8(4), 393–405 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Vabishchevich, P., Verakhovskij, V.: Difference schemes for component-wise splitting-decomposition of a domain. Mosc. Univ. Comput. Math. Cybern. 1994(3), 7–11 (1994)Google Scholar
  34. 34.
    Yanenko, N.: The method of fractional steps. The Solution of Problems of Mathematical Physics in Several Variables, VIII, 160 p. with 15 fig. Springer, Berlin-Heidelberg-New York (1971)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Nuclear Safety Institute of RASMoscowRussia
  2. 2.North-Eastern Federal UniversityYakutskRussia

Personalised recommendations