The fourth order accuracy decomposition scheme for an evolution problem

 Access to full text

 Full (PDF)

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