The fourth order accuracy decomposition scheme for an evolution problem

 Access to full text

 Full (PDF)

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.  
2. G.A. Baker, An implicit, numerical method for solving the two-dimensional heat equation. Quart. Appl. Math.17 (1959/1960) 440–443.  
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.  
4. G. Birkhoff and R.S. Varga, Implicit alternating direction methods . Trans. Amer. Math. Soc.92 (1959) 13–24.  
5. G. Birkhoff, R.S. Varga and D. Young, Alternating direction implicit methods . Adv. Comput. Academic Press, New York 3 (1962) 189–273.  
6. P.R. Chernoff, Note on product formulas for operators semigroups . J. Functional Anal.2 (1968) 238–242.  
7. P.R. Chernoff, Semigroup product formulas and addition of unbounded operators . Bull. Amer. Mat. Soc.76 (1970) 395–398.  
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.  
9. E.G. Diakonov, Difference schemes with a splitting operator for nonstationary equations . Dokl. Akad. Nauk SSSR144 (1962) 29–32.  
10. E.G. Diakonov, Difference schemes with splitting operator for higher-dimensional non-stationary problems . SSSR Comput. Math. Math. Phys.2 (1962) 549–568.  
11. J. Douglas, On numerical integration of by impilicit methods . SIAM9 (1955) 42–65.  
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.  
13. M. Dryja, Stability in W22 of schemes with splitting operators . SSSR. Comput. Math. Math. Phys.7 (1967) 296–302.  
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.  
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 Mathematics14 (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 Mathematics16 (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.  
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: M2AN36 (2002) 693–704.  
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 GSSR3 (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.  
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 GSSR60 (1970) 537–540.  
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.  
24. A.R. Gourlay and A.R. Mitchell, Intermediate boundary corrections for split operator methods in three dimensions . Nordisk Tidskr. Informations-Behandling7 (1967) 31–38.  
25. N.N. Ianenko, On Economic Implicit Schemes (Fractional steps method) . Dokl. Akad. Nauk SSSR134 (1960) 84–86.  
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.  
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.  
30. V.P. Ilin, On the splitting of difference parabolic and elliptic equations . Sibirsk. Mat. Zh6 (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 SSSR147 (1962) 25–27.  
34. S.G. Krein, Linear equations in Banach space . Nauka (1971).  
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 USSR275 (1984) 297–301.  
36. G.I. Marchuk, Split methods . Nauka (1988).  
37. G.I. Marchuk and N.N. Ianenko, The solution of a multi-dimensional kinetic equation by the splitting method . Dokl. Akad. Nauk SSSR157 (1964) 1291–1292.  
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.  
39. D. Peaceman and H. Rachford, The numerical solution of parabolic and elliptic differential equations . SIAM3 (1955) 28–41.  
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).  
41. J.L. Rogava, On the error estimation of Trotter type formulas in the case of self-Andjoint operator . Functional analysis and its aplication27 (1993) 84–86.  
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.  
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.  
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.  
46. A.A. Samarskii, P.N. Vabishchevich, Additive schemes for mathematical physics problems . Nauka (1999).  
47. Q. Sheng, Solving linear partial differential equation by exponential spliting . IMA J. Numerical Anal.9 (1989) 199–212.  
48. R. Temam, Sur la stabilité et la convergence de la méthode des pas fractionnaires . Ann. Mat. Pura Appl.4 (1968) 191–379.  
49. H. Trotter, On the product of semigroup of operators . Proc. Amer. Mat. Soc.10 (1959) 545–551.