Abstract
A purely implicit three-layer semi-discrete scheme of second-order approximation is considered in the Banach space. The three-layer semi-discrete scheme is reduced by means of the disturbance algorithm to two two-layer schemes. Using the solutions of these schemes, we construct an approximate solution of the initial problem. The stability of the constructed scheme is proved.
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References
V. I. Agoshkov and D. V. Gulua, “A perturbation algorithm for realization of finite-dimensional realization problems,” Preprint No. 253, Moscow (1990).
H. A. Alibekov and P. E. Sobolevski, “The stability of difference schemes for parabolic equations,” Dokl. Akad. Nauk SSSR, 232, No. 4, 737–740 (1977).
M. Crouzeix, “Une méthode multipas implicite-explicite pour l’approximation des équations d’ évolution paraboliques,” Numer. Math., 35, No. 3, 257–276 (1980).
M. Crouzeix and P.-A. Raviart, “Approximation des équations d’évolution linéaires par des méthodes à pas multiples,” C. R. Acad. Sci. Paris, Sér. A-B, 28, No. 6, Aiv, A367–A370 (1976).
N. Dunford and J. T. Schwartz, Linear Operators. Part 1. General Theory, Wiley, New York (1988).
S. K. Godunov and V. S. Ryabenki, Difference Schemes [in Russian], Nauka, Moscow (1973).
G. I. Marchuk, Methods of Computational Mathematics [in Russian], Nauka, Moscow (1977).
G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Methods in Nonlinear Problems of Mathematical Physics [in Russian], Nauka, Moscow (1993).
G. I. Marchuk and V. V. Shaidurov, Difference Methods and Their Extrapolations, Appl. Math., 19, Springer-Verlag, New York (1983).
A. E. Polichka and P. E. Sobolevski, “On the Rothe method of approximate solution of the Cauchy problem for a differential equation in the Banach space with a variable unbounded operator,” Differ. Uravn., 12, 1693–1704 (1976).
V. Pereyra, “On improving an approximate solution of a functional equation by deferred corrections,” Numer. Math., 8, 376–391 (1966).
V. Pereyra, “Accelerating the convergence of discretization algorithms,” SIAM J. Numer. Anal., 4, 508–533 (1967).
R. D. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Interscience, New York–London–Sidney (1967).
J. L. Rogava, Semidiscrete Schemes for Operator Differential Equations [in Russian], Tbilisi (1995).
J. Rogava and M. Tsiklauri, Higher-Order Accuracy Decomposition Schemes for Evolution Problems, Lect. Notes TICMI, 7 (2006).
M.-N. Le Roux, “Semidiscretization in time for parabolic problems,” Math. Comp., 33, No. 147, 919–931 (1979).
A. A. Samarski, Theory of Difference Schemes [in Russian], Nauka, Moscow (1977).
N. N. Yanenko, A Method of Fractional Steps of Solution of Multi-Dimensional Problems of Mathematical Physics [in Russian], Nauka, Novosibirsk (1967).
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 97, Proceedings of the International Conference “Lie Groups, Differential Equations, and Geometry,” June 10–22, 2013, Batumi, Georgia, Part 2, 2015.
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Gulua, D. On The Stability of the Disturbance Algorithm for a Semi-Discrete Scheme of Solution of an Evolutionary Equation in the Banach Space. J Math Sci 218, 770–777 (2016). https://doi.org/10.1007/s10958-016-3063-z
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DOI: https://doi.org/10.1007/s10958-016-3063-z