The fourth order accuracy decomposition scheme for an evolution problem

Zurab Gegechkori; Jemal Rogava; Mikheil Tsiklauri

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2004)

  • Volume: 38, Issue: 4, page 707-722
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Gegechkori, Zurab, Rogava, Jemal, and Tsiklauri, Mikheil. "The fourth order accuracy decomposition scheme for an evolution problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.4 (2004): 707-722. <http://eudml.org/doc/244633>.

@article{Gegechkori2004,
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.},
author = {Gegechkori, Zurab, Rogava, Jemal, Tsiklauri, Mikheil},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {decomposition method; semigroup; operator split method; Trotter formula; Cauchy abstract problem; decomposition scheme; Lie Trotter method; Lie Chernoff method; abstract evolution equation; abstract Cauchy problem},
language = {eng},
number = {4},
pages = {707-722},
publisher = {EDP-Sciences},
title = {The fourth order accuracy decomposition scheme for an evolution problem},
url = {http://eudml.org/doc/244633},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Gegechkori, Zurab
AU - Rogava, Jemal
AU - Tsiklauri, Mikheil
TI - The fourth order accuracy decomposition scheme for an evolution problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 4
SP - 707
EP - 722
AB - 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.
LA - eng
KW - decomposition method; semigroup; operator split method; Trotter formula; Cauchy abstract problem; decomposition scheme; Lie Trotter method; Lie Chernoff method; abstract evolution equation; abstract Cauchy problem
UR - http://eudml.org/doc/244633
ER -

References

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  1. [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. Zbl0183.44702
  2. [2] G.A. Baker, An implicit, numerical method for solving the two-dimensional heat equation. Quart. Appl. Math. 17 (1959/1960) 440–443. Zbl0092.32902
  3. [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. Zbl0092.32901
  4. [4] G. Birkhoff and R.S. Varga, Implicit alternating direction methods. Trans. Amer. Math. Soc. 92 (1959) 13–24. Zbl0093.31201
  5. [5] G. Birkhoff, R.S. Varga and D. Young, Alternating direction implicit methods. Adv. Comput. Academic Press, New York 3 (1962) 189–273. Zbl0111.31402
  6. [6] P.R. Chernoff, Note on product formulas for operators semigroups. J. Functional Anal. 2 (1968) 238–242. Zbl0157.21501
  7. [7] P.R. Chernoff, Semigroup product formulas and addition of unbounded operators. Bull. Amer. Mat. Soc. 76 (1970) 395–398. Zbl0193.42403
  8. [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. Zbl0853.47024
  9. [9] E.G. Diakonov, Difference schemes with a splitting operator for nonstationary equations. Dokl. Akad. Nauk SSSR 144 (1962) 29–32. Zbl0178.52001
  10. [10] E.G. Diakonov, Difference schemes with splitting operator for higher-dimensional non-stationary problems. SSSR Comput. Math. Math. Phys. 2 (1962) 549–568. Zbl0208.42302
  11. [11] J. Douglas, On numerical integration of by impilicit methods. SIAM 9 (1955) 42–65. Zbl0067.35802
  12. [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. Zbl0070.35401
  13. [13] M. Dryja, Stability in W 2 2 of schemes with splitting operators. SSSR. Comput. Math. Math. Phys. 7 (1967) 296–302. Zbl0183.44802
  14. [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. Zbl0148.39403
  15. [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. [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. Zbl1229.65092
  17. [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. Zbl1229.65094
  18. [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. Zbl0978.65088
  19. [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. Zbl1070.65562
  20. [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. [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. Zbl0432.73007
  22. [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. Zbl0222.35042
  23. [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. Zbl0278.35049
  24. [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. Zbl0235.65068
  25. [25] N.N. Ianenko, On Economic Implicit Schemes (Fractional steps method). Dokl. Akad. Nauk SSSR 134 (1960) 84–86. Zbl0099.33502
  26. [26] N.N. Ianenko, Fractional steps method of solving for multi-dimensional problems of mathematical physics. Novosibirsk, Nauka (1967). 
  27. [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. [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. Zbl0917.47041
  29. [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. Zbl0996.47046
  30. [30] V.P. Ilin, On the splitting of difference parabolic and elliptic equations. Sibirsk. Mat. Zh 6 (1965) 1425–1428. 
  31. [31] K. Iosida, Functional analysis. Springer-Verlag (1965). 
  32. [32] T. Kato, The theory of perturbations of linear operators. Mir (1972). 
  33. [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. Zbl0156.33103
  34. [34] S.G. Krein, Linear equations in Banach space. Nauka (1971). MR374949
  35. [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. Zbl0601.65047
  36. [36] G.I. Marchuk, Split methods. Nauka (1988). Zbl0653.65065MR986974
  37. [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. Zbl0163.23201
  38. [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. Zbl0162.47403
  39. [39] D. Peaceman and H. Rachford, The numerical solution of parabolic and elliptic differential equations. SIAM 3 (1955) 28–41. Zbl0067.35801
  40. [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). Zbl0308.47002MR493420
  41. [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. Zbl0814.47050
  42. [42] J.L. Rogava, Semi-discrete schemes for operator differential equations. Tbilisi, Georgian Technical University press (1995). 
  43. [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. Zbl0273.65078
  44. [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. Zbl0143.17503
  45. [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. Zbl0273.65080
  46. [46] A.A. Samarskii, P.N. Vabishchevich, Additive schemes for mathematical physics problems. Nauka (1999). Zbl0963.65091MR1788271
  47. [47] Q. Sheng, Solving linear partial differential equation by exponential spliting. IMA J. Numerical Anal. 9 (1989) 199–212. Zbl0676.65116
  48. [48] R. Temam, Sur la stabilité et la convergence de la méthode des pas fractionnaires. Ann. Mat. Pura Appl. 4 (1968) 191–379. Zbl0174.45804
  49. [49] H. Trotter, On the product of semigroup of operators. Proc. Amer. Mat. Soc. 10 (1959) 545–551. Zbl0099.10401

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