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

DOI:
https://doi.org/10.1090/S0025-5718-2011-02492-0

Published electronically:
June 20, 2011

MathSciNet review:
2833495

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- V. N. Abrashin,
*A variant of the method of variable directions for the solution of multidimensional problems in mathematical physics. I*, Differentsial′nye Uravneniya**26**(1990), no. 2, 314–323, 366 (Russian); English transl., Differential Equations**26**(1990), no. 2, 243–250. MR**1050397** - Jim Douglas Jr. and H. H. Rachford Jr.,
*On the numerical solution of heat conduction problems in two and three space variables*, Trans. Amer. Math. Soc.**82**(1956), 421–439. MR**84194**, DOI https://doi.org/10.1090/S0002-9947-1956-0084194-4 - Richard E. Ewing,
*Numerical solution of Sobolev partial differential equations*, SIAM J. Numer. Anal.**12**(1975), 345–363. MR**395265**, DOI https://doi.org/10.1137/0712028 - Richard E. Ewing,
*Time-stepping Galerkin methods for nonlinear Sobolev partial differential equations*, SIAM J. Numer. Anal.**15**(1978), no. 6, 1125–1150. MR**512687**, DOI https://doi.org/10.1137/0715075 - D. G. Gordeziani and G. V. Meladze,
*The simulation of the third boundary value problem for multidimensional parabolic equations in an arbitrary domain by one-dimensional equations*, Ž. Vyčisl. Mat i Mat. Fiz.**14**(1974), 246–250, 271 (Russian). MR**400734** - Tarek P. A. Mathew,
*Domain decomposition methods for the numerical solution of partial differential equations*, Lecture Notes in Computational Science and Engineering, vol. 61, Springer-Verlag, Berlin, 2008. MR**2445659** - G. I. Marchuk,
*Splitting and alternating direction methods*, Handbook of numerical analysis, Vol. I, Handb. Numer. Anal., I, North-Holland, Amsterdam, 1990, pp. 197–462. MR**1039325** - D. W. Peaceman and H. H. Rachford Jr.,
*The numerical solution of parabolic and elliptic differential equations*, J. Soc. Indust. Appl. Math.**3**(1955), 28–41. MR**71874** - A.A. Samarskii,
*Regularization of difference schemes*, Computational Mathematics and Mathematical Physics**7**(1967), no. 1, 79–120. - Alexander A. Samarskii,
*The theory of difference schemes*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 240, Marcel Dekker, Inc., New York, 2001. MR**1818323** - A. A. Samarskiĭ and A. V. Gulin,
*Ustoĭ chivost′raznostnykh skhem*, 2nd ed., Èditorial URSS, Moscow, 2004 (Russian, with Russian summary). MR**2263771** - A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich,
*Difference schemes with operator factors*, Mathematics and its Applications, vol. 546, Kluwer Academic Publishers, Dordrecht, 2002. MR**1950844** - A.A. Samarskii, P.P. Matus, and P.N. Vabishchevich,
*Stability of vector additive schemes*, Doklady Mathematics**58**(1998), no. 1, 133–135. - A. A. Samarskiĭ, Petr N. Vabishchevich, and Lubin G. Vulkov (eds.),
*Finite difference methods: theory and applications*, Nova Science Publishers, Inc., Commack, NY, 1999. MR**1714030** - A.A. Samarskii and P.N. Vabishchevich,
*Regularized additive full approximation schemes*, Doklady Mathematics**57**(1998), 83–86. - S.L. Sobolev,
*Some new problems in mathematical physics*, Izv. Akad. Nauk SSSR Ser. Mat**18**(1954), 3–50. - R. E. Showalter,
*Partial differential equations of Sobolev-Galpern type*, Pacific J. Math.**31**(1969), 787–793. MR**252870** - R. E. Showalter and T. W. Ting,
*Pseudoparabolic partial differential equations*, SIAM J. Math. Anal.**1**(1970), 1–26. MR**437936**, DOI https://doi.org/10.1137/0501001 - 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). - P. N. Vabishchevich,
*Vector additive difference schemes for first-order evolution equations*, Zh. Vychisl. Mat. i Mat. Fiz.**36**(1996), no. 3, 44–51 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys.**36**(1996), no. 3, 317–322. MR**1382641** - P. N. Vabishchevich,
*Regularized additive operator-difference schemes*, Zh. Vychisl. Mat. Mat. Fiz.**50**(2010), no. 3, 449–457 (Russian, with Russian summary); English transl., Comput. Math. Math. Phys.**50**(2010), no. 3, 428–436. MR**2681922**, DOI https://doi.org/10.1134/S096554251003005X - P. N. Vabishchevich,
*Domain decomposition methods with overlapping subdomains for the time-dependent problems of mathematical physics*, Comput. Methods Appl. Math.**8**(2008), no. 4, 393–405. MR**2604752**, DOI https://doi.org/10.2478/cmam-2008-0029 - N. N. Yanenko,
*The method of fractional steps. The solution of problems of mathematical physics in several variables*, Springer-Verlag, New York-Heidelberg, 1971. Translated from the Russian by T. Cheron. English translation edited by M. Holt. MR**0307493**

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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

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.