Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-22T23:37:13.511Z Has data issue: false hasContentIssue false
  • English
  • Français

The fourth order accuracy decomposition schemefor an evolution problem

Published online by Cambridge University Press:  15 August 2004

Zurab Gegechkori ,
Jemal Rogava  and
Mikheil Tsiklauri
Zurab Gegechkori
Affiliation:
I. Vekua Institute of Applied Mathematics, University Str. 2, 0143, Tbilisi, Georgia. [email protected].
Jemal Rogava
Affiliation:
I. Vekua Institute of Applied Mathematics, University Str. 2, 0143, Tbilisi, Georgia. [email protected].
Mikheil Tsiklauri
Affiliation:
I. Vekua Institute of Applied Mathematics, University Str. 2, 0143, Tbilisi, Georgia. [email protected].
Get access

Abstract

In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.

Keywords

Decomposition methodsemigroupoperator split methodTrotter formulaCauchy abstract problem.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 4 , July 2004 , pp. 707 - 722
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andreev, V.B., On difference schemes with a splitting operator for general p-dimensional parabolic equations of second order with mixed derivatives . SSSR Comput. Math. Math. Phys. 7 (1967) 312–321.
G.A. Baker, An implicit, numerical method for solving the two-dimensional heat equation. Quart. Appl. Math. 17 (1959/1960) 440–443.
G.A. Baker and T.A. Oliphant, An implicit, numerical method for solving the two-dimensional heat equation . Quart. Appl. Math. 17 (1959/1960) 361–373.
Birkhoff, G. and Varga, R.S., Implicit alternating direction methods . Trans. Amer. Math. Soc. 92 (1959) 13–24. CrossRef
G. Birkhoff, R.S. Varga and D. Young, Alternating direction implicit methods . Adv. Comput. Academic Press, New York 3 (1962) 189–273.
Chernoff, P.R., Note on product formulas for operators semigroups . J. Functional Anal. 2 (1968) 238–242. CrossRef
Chernoff, P.R., Semigroup product formulas and addition of unbounded operators . Bull. Amer. Mat. Soc. 76 (1970) 395–398. CrossRef
Dia, B.O. and Schatzman, M., Comutateurs semi-groupes holomorphes et applications aux directions alternées . RAIRO Modél. Math. Anal. Numér. 30 (1996) 343–383. CrossRef
Diakonov, E.G., Difference schemes with a splitting operator for nonstationary equations . Dokl. Akad. Nauk SSSR 144 (1962) 29–32.
Diakonov, E.G., Difference schemes with splitting operator for higher-dimensional non-stationary problems . SSSR Comput. Math. Math. Phys. 2 (1962) 549–568.
Douglas, J., On numerical integration of by impilicit methods . SIAM 9 (1955) 42–65.
Douglas, J. and Rachford, H., On the numerical solution of heat condition problems in two and three space variables . Trans. Amer. Math. Soc. 82 (1956) 421–439. CrossRef
Dryja, M., Stability in W 2 2 of schemes with splitting operators . SSSR. Comput. Math. Math. Phys. 7 (1967) 296–302.
Fairweather, G., Gourlay, A.R. and Mitchell, A.R., Some high accuracy difference schemes with a splitting operator for equations of parabolic and elliptic type . Numer. Math. 10 (1967) 56–66. CrossRef
Fryazinov, I.V., Increased precision order economical schemes for the solution of parabolic type multi-dimensional equations . SSSR. Comput. Math. Math. Phys. 9 (1969) 1319–1326.
Z.G. Gegechkori, J.L. Rogava and M.A. Tsiklauri, High-degree precision decomposition method for an evolution problem . Tbilisi, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 14 (1999) 45–48.
Z.G. Gegechkori, J.L. Rogava and M.A. Tsiklauri, High degree precision decomposition formulas of semigroup approximation . Tbilisi, Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 16 (2001) 89–92.
Z.G. Gegechkori, J.L. Rogava and M.A. Tsiklauri, Sequention-Parallel method of high degree precision for Cauchy abstract problem solution. Minsk, Comput. Methods in Appl. Math. 1 (2001) 173–187.
Gegechkori, Z.G., Rogava, J.L. and Tsiklauri, M.A., High degree precision decomposition method for the evolution problem with an operator under a split form . ESAIM: M2AN 36 (2002) 693–704. CrossRef
D.G. Gordeziani, On application of local one-dimensional method for solving parabolic type multi-dimensional problems of 2m-degree, Proc. of Science Academy of GSSR 3 (1965) 535–542.
D.G. Gordeziani and A.A. Samarskii, Some problems of plates and shells thermo elasticity and method of summary approximation . Complex analysis and it's applications (1978) 173–186.
D.G. Gordeziani and H.V. Meladze, On modeling multi-dimensional quasi-linear equation of parabolic type by one-dimensional ones, Proc. of Science Academy of GSSR 60 (1970) 537–540.
Gordeziani, D.G. and Meladze, H.V., On modeling of third boundary value problem for the multi-dimensional parabolic equations of arbitrary area by the one-dimensional equations . SSSR Comput. Math. Math. Phys. 14 (1974) 246–250.
Gourlay, A.R. and Mitchell, A.R., Intermediate boundary corrections for split operator methods in three dimensions . Nordisk Tidskr. Informations-Behandling 7 (1967) 31–38.
Ianenko, N.N., Economic Implicit Schemes (Fractional, On steps method) . Dokl. Akad. Nauk SSSR 134 (1960) 84–86.
N.N. Ianenko, Fractional steps method of solving for multi-dimensional problems of mathematical physics . Novosibirsk, Nauka (1967).
N.N. Ianenko and G.V. Demidov, The method of weak approximation as a constructive method for building up a solution of the Cauchy problem . Izdat. “Nauka”, Sibirsk. Otdel., Novosibirsk. Certain Problems Numer. Appl. Math. (1966) 60–83.
Ichinose, T. and Takanobu, S., The norm estimate of the difference between the Kac operator and the Schrodinger emigroup . Nagoya Math. J. 149 (1998) 53–81. CrossRef
Ichinose, T. and Tamura, H., The norm convergence of the Trotter-Kato product formula with error bound . Commun. Math. Phys. 217 (2001) 489–502. CrossRef
Ilin, V.P., On the splitting of difference parabolic and elliptic equations . Sibirsk. Mat. Zh 6 (1965) 1425–1428.
K. Iosida, Functional analysis . Springer-Verlag (1965).
T. Kato, The theory of perturbations of linear operators . Mir (1972).
Konovalov, A.N., The fractional step method for solving the Cauchy problem for an n-dimensional oscillation equation . Dokl. Akad. Nauk SSSR 147 (1962) 25–27.
S.G. Krein, Linear equations in Banach space . Nauka (1971).
Kuzyk, A.M. and Makarov, V.L., Estimation of an exactitude of summarized approximation of a solution of Cauchy abstract problem . Dokl. Akad. Nauk USSR 275 (1984) 297–301.
G.I. Marchuk, Split methods . Nauka (1988).
Marchuk, G.I. and Ianenko, N.N., The solution of a multi-dimensional kinetic equation by the splitting method . Dokl. Akad. Nauk SSSR 157 (1964) 1291–1292.
Marchuk, G.I. and Sultangazin, U.M., On a proof of the splitting method for the equation of radiation transfer . SSSR. Comput. Math. Math. Phys. 5 (1965) 852–863.
Peaceman, D. and Rachford, H., The numerical solution of parabolic and elliptic differential equations . SIAM 3 (1955) 28–41.
M. Reed and B. Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness . New York-London, Academic Press [Harcourt Brace Jovanovich, Publishers] (1975).
Rogava, J.L., On the error estimation of Trotter type formulas in the case of self-Andjoint operator . Functional analysis and its aplication 27 (1993) 84–86.
J.L. Rogava, Semi-discrete schemes for operator differential equations . Tbilisi, Georgian Technical University press (1995).
Samarskii, A.A., On an economical difference method for the solution of a multi-dimensional parabolic equation in an arbitrary region . SSSR Comput. Math. Math. Phys. 2 (1962) 787–811.
Samarskii, A.A., On the convergence of the method of fractional steps for the heat equation . SSSR Comput. Math. Math. Phys. 2 (1962) 1117–1121.
Samarskii, A.A., Locally homogeneous difference schemes for higher-dimensional equations of hyperbolic type in an arbitrary region. SSSR Comput. Math. Math. Phys. 4 (1962) 638–648.
A.A. Samarskii, P.N. Vabishchevich, Additive schemes for mathematical physics problems . Nauka (1999).
Sheng, Q., Solving linear partial differential equation by exponential spliting . IMA J. Numerical Anal. 9 (1989) 199–212. CrossRef
Temam, R., Sur la stabilité et la convergence de la méthode des pas fractionnaires . Ann. Mat. Pura Appl. 4 (1968) 191–379. CrossRef
Trotter, H., On the product of semigroup of operators . Proc. Amer. Mat. Soc. 10 (1959) 545–551. CrossRef