Abstract
Nonstationary problems are solved numerically by applying multilevel (with more than two levels) time approximations. They are easy to construct and relatively easy to study in the case of uniform grids. However, the numerical study of application-oriented problems often involves approximations with a variable time step. The construction of multilevel schemes on nonuniform grids is associated with maintaining the prescribed accuracy and ensuring the stability of the approximate solution. In this paper, three-level schemes for the approximate solution of the Cauchy problem for a second-order evolution equation are constructed in the special case of a doubled or halved step size. The focus is on the approximation features in the transition between different step sizes. The study is based on general results of the stability (well-posedness) theory of operator-difference schemes in a finite-dimensional Hilbert space. Estimates for stability with respect to initial data and the right-hand side are obtained in the case of a doubled or halved time step.
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REFERENCES
E. Hairer and G. Wanner, Solving Ordinary Differential Equations. II: Stiff and Differential-Algebraic Problems (Springer, Berlin, 1996).
R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (SIAM, Philadelphia, 2007).
P. N. Vabishchevich, Numerical Methods for Solving Nonstationary Problems (LENAND, Moscow, 2021) [in Russian].
A. A. Samarskii, The Theory of Difference Schemes (Marcel Dekker, New York, 2001).
A. A. Samarskii and A. V. Gulin, Stability of Finite Difference Schemes (Nauka, Moscow, 1973) [in Russian].
A. A. Samarskii, P. P. Matus, and P. N. Vabishchevich, Difference Schemes with Operator Factors (Kluwer, Dordrecht, 2002).
A. A. Samarskii, P. N. Vabishchevich, E. L. Makarevich, and P. P. Matus, “Stability of three-layer difference schemes on time-nonuniform grids,” Dokl. Math. 63 (1), 106–108 (2001).
P. Matus and E. Zyuzina, “Three-level difference schemes on non-uniform in time grids,” Comput. Methods Appl. Math. 1 (3), 265–284 (2001).
Funding
This work was supported by the Russian Science Foundation, project no. 23-41-00037 (Section 2) and project no. 23-71-30013 (Section 3).
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Translated by I. Ruzanova
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Vabishchevich, P.N. Three-Level Schemes with Double Change in the Time Step. Comput. Math. and Math. Phys. 63, 1989–1995 (2023). https://doi.org/10.1134/S0965542523110192
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DOI: https://doi.org/10.1134/S0965542523110192