The fourth order accuracy decomposition scheme for an evolution problem

Gegechkori, Zurab; Rogava, Jemal; Tsiklauri, Mikheil

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 4, pp. 707-722.

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.

Classification : 65M12, 65M15, 65M55
Mots clés : decomposition method, semigroup, operator split method, Trotter formula, Cauchy abstract problem

[1] V.B. Andreev, 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. | MR | Zbl

[2] G.A. Baker, An implicit, numerical method for solving the two-dimensional heat equation. Quart. Appl. Math. 17 (1959/1960) 440-443. | MR | Zbl

[3] 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. | MR | Zbl

[4] G. Birkhoff and R.S. Varga, Implicit alternating direction methods. Trans. Amer. Math. Soc. 92 (1959) 13-24. | MR | Zbl

[5] G. Birkhoff, R.S. Varga and D. Young, Alternating direction implicit methods. Adv. Comput. Academic Press, New York 3 (1962) 189-273. | MR | Zbl

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

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

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

[9] E.G. Diakonov, Difference schemes with a splitting operator for nonstationary equations. Dokl. Akad. Nauk SSSR 144 (1962) 29-32. | MR | Zbl

[10] E.G. Diakonov, Difference schemes with splitting operator for higher-dimensional non-stationary problems. SSSR Comput. Math. Math. Phys. 2 (1962) 549-568. | Zbl

[11] J. Douglas, On numerical integration of by impilicit methods. SIAM 9 (1955) 42-65. | Zbl

[12] J. Douglas and H. Rachford, On the numerical solution of heat condition problems in two and three space variables. Trans. Amer. Math. Soc. 82 (1956) 421-439. | Zbl

[13] M. Dryja, Stability in $W_2{}^2$ of schemes with splitting operators. SSSR. Comput. Math. Math. Phys. 7 (1967) 296-302. | Zbl

[14] G. Fairweather, A.R. Gourlay and A.R. Mitchell, Some high accuracy difference schemes with a splitting operator for equations of parabolic and elliptic type. Numer. Math. 10 (1967) 56-66. | Zbl

[15] I.V. Fryazinov, Increased precision order economical schemes for the solution of parabolic type multi-dimensional equations. SSSR. Comput. Math. Math. Phys. 9 (1969) 1319-1326.

[16] 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.

[17] 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.

[18] 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. | Zbl

[19] Z.G. Gegechkori, J.L. Rogava and M.A. Tsiklauri, High degree precision decomposition method for the evolution problem with an operator under a split form. ESAIM: M2AN 36 (2002) 693-704. | Numdam | Zbl

[20] 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.

[21] 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. | Zbl

[22] 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. | Zbl

[23] D.G. Gordeziani and H.V. Meladze, 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. | Zbl

[24] A.R. Gourlay and A.R. Mitchell, Intermediate boundary corrections for split operator methods in three dimensions. Nordisk Tidskr. Informations-Behandling 7 (1967) 31-38. | Zbl

[25] N.N. Ianenko, On Economic Implicit Schemes (Fractional steps method). Dokl. Akad. Nauk SSSR 134 (1960) 84-86. | Zbl

[26] N.N. Ianenko, Fractional steps method of solving for multi-dimensional problems of mathematical physics. Novosibirsk, Nauka (1967).

[27] 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.

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

[29] T. Ichinose and H. Tamura, The norm convergence of the Trotter-Kato product formula with error bound. Commun. Math. Phys. 217 (2001) 489-502. | Zbl

[30] V.P. Ilin, On the splitting of difference parabolic and elliptic equations. Sibirsk. Mat. Zh 6 (1965) 1425-1428.

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

[32] T. Kato, The theory of perturbations of linear operators. Mir (1972).

[33] A.N. Konovalov, The fractional step method for solving the Cauchy problem for an $n$-dimensional oscillation equation. Dokl. Akad. Nauk SSSR 147 (1962) 25-27. | Zbl

[34] S.G. Krein, Linear equations in Banach space. Nauka (1971). | MR

[35] A.M. Kuzyk and V.L. Makarov, Estimation of an exactitude of summarized approximation of a solution of Cauchy abstract problem. Dokl. Akad. Nauk USSR 275 (1984) 297-301. | Zbl

[36] G.I. Marchuk, Split methods. Nauka (1988). | MR

[37] G.I. Marchuk and N.N. Ianenko, The solution of a multi-dimensional kinetic equation by the splitting method. Dokl. Akad. Nauk SSSR 157 (1964) 1291-1292. | Zbl

[38] G.I. Marchuk and U.M. Sultangazin, On a proof of the splitting method for the equation of radiation transfer. SSSR. Comput. Math. Math. Phys. 5 (1965) 852-863. | Zbl

[39] D. Peaceman and H. Rachford, The numerical solution of parabolic and elliptic differential equations. SIAM 3 (1955) 28-41. | Zbl

[40] 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). | MR | Zbl

[41] J.L. Rogava, On the error estimation of Trotter type formulas in the case of self-Andjoint operator. Functional analysis and its aplication 27 (1993) 84-86. | Zbl

[42] J.L. Rogava, Semi-discrete schemes for operator differential equations. Tbilisi, Georgian Technical University press (1995).

[43] A.A. Samarskii, 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. | Zbl

[44] A.A. Samarskii, On the convergence of the method of fractional steps for the heat equation. SSSR Comput. Math. Math. Phys. 2 (1962) 1117-1121. | Zbl

[45] A.A. Samarskii, Locally homogeneous difference schemes for higher-dimensional equations of hyperbolic type in an arbitrary region. SSSR Comput. Math. Math. Phys. 4 (1962) 638-648. | Zbl

[46] A.A. Samarskii, P.N. Vabishchevich, Additive schemes for mathematical physics problems. Nauka (1999). | MR | Zbl

[47] Q. Sheng, Solving linear partial differential equation by exponential spliting. IMA J. Numerical Anal. 9 (1989) 199-212. | Zbl

[48] R. Temam, Sur la stabilité et la convergence de la méthode des pas fractionnaires. Ann. Mat. Pura Appl. 4 (1968) 191-379. | Zbl

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