High degree precision decomposition method for the evolution problem with an operator under a split form
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 4, pp. 693-704.

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

DOI : 10.1051/m2an:2002030
Classification : 65M12, 65M15, 65M55
Mots-clés : decomposition method, semigroup, Trotter formula, Cauchy abstract problem 
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     title = {High degree precision decomposition method for the evolution problem with an operator under a split form},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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Gegechkori, Zurab; Rogava, Jemal; Tsiklauri, Mikheil. High degree precision decomposition method for the evolution problem with an operator under a split form. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 4, pp. 693-704. doi : 10.1051/m2an:2002030. http://www.numdam.org/articles/10.1051/m2an:2002030/

[1] P.R. Chernoff, Note on product formulas for operators semigroups. J. Funct. Anal. 2 (1968) 238-242. | Zbl

[2] P.R. Chernoff, Semigroup product formulas and addition of unbounded operators. Bull. Amer. Mat. Soc. 76 (1970) 395-398. | Zbl

[3] B.O. Dia and M. Schatzman, Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées. RAIRO Modél. Math. Anal. Numér. 30 (1996) 343-383. | Numdam | Zbl

[4] E.G. Diakonov, Difference schemas with decomposition operator for Multidimensional problems. JNM and MPh 2 (1962) 311-319.

[5] I.V. Fryazinov, Increased precision order economical schemas for the solution of parabolic type multidimensional equations. JNM and MPh 9 (1969) 1319-1326.

[6] Z.G. Gegechkori, J.A. Rogava and M.A Tsiklauri, Sequential-Parallel method of high degree precision for Cauchy abstract problem solution. Tbilisi, in Reports of the Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics 14 (1999). | MR

[7] D.G. Gordeziani, On application of local one dimensional method for solving parabolic type multidimensional problems of 2m-degree. Proc. Acad. Sci. GSSR 3 (1965) 535-542.

[8] D.G. Gordeziani and H.V. Meladze, On modeling multidimensional quasi-linear equation of parabolic type by one-dimensional ones. Proc. Acad. Sci. GSSR 60 (1970) 537-540. | Zbl

[9] D.G. Gordeziani and H.V. Meladze, On modeling of third boundary value problem for the multidimensional parabolic equations of an arbitrary area by one-dimensional equations. JNM and MPh 14 (1974) 246-250. | Zbl

[10] D.G. Gordeziani and A.A. Samarskii, Some problems of plates and shells thermo elasticity and method of summary approximation. Complex Anal. Appl. (1978) 173-186. | Zbl

[11] N.N. Ianenko, Fractional steps method of solving multidimensional problems of mathematical physics. Nauka, Novosibirsk (1967) 196 p.

[12] T. Ichinose and S. Takanobu, The norm estimate of the difference between the Kac operator and the Schrodinger semigroup. Nagoya Math. J. 149 (1998) 53-81. | Zbl

[13] K. Iosida, Functional analysis. Springer-Verlag (1965).

[14] T. Kato, The theory of perturbations of linear operators. Mir, Moscow (1972) 740 p. | Zbl

[15] S.G. Krein, Linear equations in Banach space. Nauka, Moscow (1971), 464 p. | Zbl

[16] A.M. Kuzyk and V.L. Makarov, Estimation of exactitude of summarized approximation of a solution of the Cauchy abstract problem. RAN USSR 275 (1984) 297-301. | Zbl

[17] G.I. Marchuk, Split methods. Nauka, Moscow (1988) 264 p. | MR | Zbl

[18] J.A. Rogava, On the error estimation of Trotter type formulas in the case of self-Adjoint operator. Funct. Anal. Appl. 27 (1993) 84-86. | Zbl

[19] J.A. Rogava, Semi-discrete schemas for operator differential equations. Tbilisi, Georgian Technical University press (1995) 288 p.

[20] A.A. Samarskii, Difference schemas theory. Nauka, Moscow (1977), 656 p. | MR | Zbl

[21] A.A. Samarskii and P.N. Vabishchevich, Additive schemas for mathematical physics problems. Nauka, Moscow (1999). | MR | Zbl

[22] R. Temam, Quelques méthodes de décomposition en analyse numérique. Actes Congrés Intern. Math. (1970) 311-319. | Zbl

[23] H. Trotter, On the product of semigroup of operators. Proc. Amer. Mat. Soc. 10 (1959) 545-551. | Zbl

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