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On a new class of additive (splitting) operator-difference schemes

Author: Petr N. Vabishchevich
Journal: Math. Comp. 81 (2012), 267-276
MSC (2010): Primary 65M06, 65M12
Published electronically: June 20, 2011
MathSciNet review: 2833495
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Abstract: Many applied time-dependent problems are characterized by an additive representation of the problem operator. Additive schemes are constructed using such a splitting and are associated with the transition to a new time level on the basis of the solution of more simple problems for the individual operators in the additive decomposition. We consider a new class of additive schemes for problems with additive representation of the operator at the time derivative. In this paper we construct and study the vector operator-difference schemes, which are characterized by a transition from the single initial evolution equation to a system of evolution equations.

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  • [1] V.N. Abrashin, A variant of the method of variable directions for the solution of multi- dimensional problems of mathematical-physics, Differ. Equations 26 (1990), no. 2, 243-250. MR 1050397 (91c:35003)
  • [2] J. Douglas and H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Trans. Amer. Math. Soc. 82 (1956), 421-439. MR 0084194 (18:827f)
  • [3] R.E. Ewing, Numerical solution of Sobolev partial differential equations, SIAM Journal on Numerical Analysis 12 (1975), no. 3, 345-363. MR 0395265 (52:16062)
  • [4] R.E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM Journal on Numerical Analysis 15 (1978), no. 6, 1125-1150. MR 512687 (80b:65136)
  • [5] D.G. Gordeziani and G.V. Meladze, Simulation of the third boundary value problem for multidimensional parabolic equations in an arbitrary domain by one-dimensional equations, Computational Mathematics and Mathematical Physics 14 (1974), no. 1, 249-253. MR 0400734 (53:4564)
  • [6] T. Mathew, Domain decomposition methods for the numerical solution of partial differential equations, Lecture Notes in Computational Science and Engineering 61. Berlin: Springer. xiv+764 pp., 2008. MR 2445659 (2010b:65006)
  • [7] G.I. Marchuk, Splitting and alternating direction methods, Handbook of numerical analysis, Vol. I, 1990, 197-462.
  • [8] D.W. Peaceman and H.H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3 (1955), 28-41. MR 0071874 (17:196d)
  • [9] A.A. Samarskii, Regularization of difference schemes, Computational Mathematics and Mathematical Physics 7 (1967), no. 1, 79-120.
  • [10] A.A. Samarskii, The theory of difference schemes, Pure and Applied Mathematics, 240. Marcel Dekker, Inc., New York, xviii+761 pp. MR 1818323 (2002c:65003)
  • [11] A.A. Samarskii and A.V. Gulin, Stability of difference schemes, Moscow: URSS. 384 p., 2005 (Russian). MR 2263771 (2008a:65001)
  • [12] A.A. Samarskii, P.P. Matus, and P.N. Vabishchevich, Difference schemes with operator factors, Mathematics and its Applications (Dordrecht). 546. Dordrecht: Kluwer Academic Publishers. x+384 pp., 2002. MR 1950844 (2003k:65095)
  • [13] A.A. Samarskii, P.P. Matus, and P.N. Vabishchevich, Stability of vector additive schemes, Doklady Mathematics 58 (1998), no. 1, 133-135.
  • [14] A.A. Samarskii and P.N. Vabishchevich, Additive schemes for problems of mathematical physics, Moscow: Nauka. 320 p., 1999 (Russian). MR 1714030 (2000e:65004)
  • [15] A.A. Samarskii and P.N. Vabishchevich, Regularized additive full approximation schemes, Doklady Mathematics 57 (1998), 83-86.
  • [16] S.L. Sobolev, Some new problems in mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat 18 (1954), 3-50.
  • [17] R.E. Showalter, Partial differential equations of Sobolev-Galpern type, Pacific J. Math 31 (1969), no. 3, 787-793. MR 0252870 (40:6085)
  • [18] R.E. Showalter and T.W. Ting, Pseudoparabolic partial differential equations, Siam J. Math. Anal 1 (1970), no. 1, 1-26. MR 0437936 (55:10857)
  • [19] A.G. Sveshnikov, A.B. Al'shin, M.O. Korpusov, and Y.D. Pletner, Linear and nonlinear equations of Sobolev type, Moscow: Fizmatlit. 734 p., 2007 (Russian).
  • [20] P.N. Vabishchevich, Vector additive difference schemes for first-order evolution equations, Computational Mathematics and Mathematical Physics 36 (1996), no. 3, 317-322. MR 1382641
  • [21] P.N. Vabishchevich, Regularized additive operator-difference schemes, Computational Mathematics and Mathematical Physics 50 (2010), no. 3, 428-436. MR 2681922 (2011f:65134)
  • [22] P.N. Vabishchevich, Domain decomposition methods with overlapping subdomains for the time-dependent problems of mathematical physics, Computational Methods in Applied Mathematics 8 (2008), 393-405. MR 2604752
  • [23] N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables, Berlin-Heidelberg-New York: Springer Verlag, VIII, 160 p., 1971. MR 0307493 (46:6613)

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Additional Information

Petr N. Vabishchevich
Affiliation: Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 4 Miusskaya Sq., 125047 Moscow, Russia

Keywords: Evolutionary problems, splitting schemes, the stability of operator-difference schemes, vector additive schemes
Received by editor(s): May 12, 2010
Received by editor(s) in revised form: September 7, 2010
Published electronically: June 20, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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