On a new class of additive (splitting) operatordifference schemes
Author:
Petr N. Vabishchevich
Journal:
Math. Comp. 81 (2012), 267276
MSC (2010):
Primary 65M06, 65M12
Published electronically:
June 20, 2011
MathSciNet review:
2833495
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Additional Information
Abstract: Many applied timedependent 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 operatordifference schemes, which are characterized by a transition from the single initial evolution equation to a system of evolution equations.
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 V.N. Abrashin, A variant of the method of variable directions for the solution of multi dimensional problems of mathematicalphysics, Differ. Equations 26 (1990), no. 2, 243250. MR 1050397 (91c:35003)
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 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), 421439. MR 0084194 (18:827f)
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 R.E. Ewing, Numerical solution of Sobolev partial differential equations, SIAM Journal on Numerical Analysis 12 (1975), no. 3, 345363. MR 0395265 (52:16062)
 [4]
 R.E. Ewing, Timestepping Galerkin methods for nonlinear Sobolev partial differential equations, SIAM Journal on Numerical Analysis 15 (1978), no. 6, 11251150. MR 512687 (80b:65136)
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 D.G. Gordeziani and G.V. Meladze, Simulation of the third boundary value problem for multidimensional parabolic equations in an arbitrary domain by onedimensional equations, Computational Mathematics and Mathematical Physics 14 (1974), no. 1, 249253. MR 0400734 (53:4564)
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 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)
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 G.I. Marchuk, Splitting and alternating direction methods, Handbook of numerical analysis, Vol. I, 1990, 197462.
 [8]
 D.W. Peaceman and H.H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. Soc. Ind. Appl. Math. 3 (1955), 2841. MR 0071874 (17:196d)
 [9]
 A.A. Samarskii, Regularization of difference schemes, Computational Mathematics and Mathematical Physics 7 (1967), no. 1, 79120.
 [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)
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 A.A. Samarskii and A.V. Gulin, Stability of difference schemes, Moscow: URSS. 384 p., 2005 (Russian). MR 2263771 (2008a:65001)
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 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)
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 A.A. Samarskii and P.N. Vabishchevich, Additive schemes for problems of mathematical physics, Moscow: Nauka. 320 p., 1999 (Russian). MR 1714030 (2000e:65004)
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 A.A. Samarskii and P.N. Vabishchevich, Regularized additive full approximation schemes, Doklady Mathematics 57 (1998), 8386.
 [16]
 S.L. Sobolev, Some new problems in mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat 18 (1954), 350.
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 R.E. Showalter, Partial differential equations of SobolevGalpern type, Pacific J. Math 31 (1969), no. 3, 787793. MR 0252870 (40:6085)
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 P.N. Vabishchevich, Vector additive difference schemes for firstorder evolution equations, Computational Mathematics and Mathematical Physics 36 (1996), no. 3, 317322. MR 1382641
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 P.N. Vabishchevich, Regularized additive operatordifference schemes, Computational Mathematics and Mathematical Physics 50 (2010), no. 3, 428436. MR 2681922 (2011f:65134)
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 P.N. Vabishchevich, Domain decomposition methods with overlapping subdomains for the timedependent problems of mathematical physics, Computational Methods in Applied Mathematics 8 (2008), 393405. MR 2604752
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 N.N. Yanenko, The method of fractional steps. The solution of problems of mathematical physics in several variables, BerlinHeidelbergNew 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
Email:
vabishchevich@gmail.com
DOI:
http://dx.doi.org/10.1090/S002557182011024920
PII:
S 00255718(2011)024920
Keywords:
Evolutionary problems,
splitting schemes,
the stability of operatordifference 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.
