Email this article SPLITTING SCHEMES FOR EVOLUTION EQUATIONS WITH FACTORIZED OPERATOR
- Authors: Vabishchevich P.N.1,2
-
Affiliations:
- Lomonosov Moscow State University
- North-Caucasus Federal University
- Issue: Vol 60, No 7 (2024)
- Pages: 876–885
- Section: NUMERICAL METHODS
- URL: https://ogarev-online.ru/0374-0641/article/view/265845
- DOI: https://doi.org/10.31857/S0374064124070024
- EDN: https://elibrary.ru/KOFBCW
- ID: 265845
Cite item
Open Access
Access granted
Subscription Access
Abstract
In the approximate solution of the Cauchy problem for evolution equations, the problem operator can often be represented as a sum of simpler operators. This makes it possible to construct operatordifference splitting schemes, when the transition to a new level in time is provided by solving problems for separate operator summands. We consider nonstationary problems, the main feature of which is related to the representation of the problem operator as a product of the operator 𝐴 by its conjugate operator 𝐴*. Based on the transformation of the original equation to a system of two equations, we construct time approximations for second-order evolutionary equations when the additive representation holds for the operator 𝐴. Unconditional stable splitting schemes are proposed, the study of which is carried out with the help of general results of the theory of stability (correctness) of operator-difference schemes in Hilbert spaces.
About the authors
P. N. Vabishchevich
Lomonosov Moscow State University; North-Caucasus Federal University
Email: vab@cs.msu.ru
Moscow, Russia; Stavropol, Russia
References
- Dautray, K. Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1. Physical Origins and Classical Methods / К. Dautray, J.-L Lions. — Berlin : Springer, 2000. — 722 p.
- Dautray, K. and Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1. Physical Origins and Classical Methods, Berlin: Springer, 2000.
- Fung, Y.C. Classical and Computational Solid Mechanics / Y.C. Fung, P. Tong, X. Chen. — New Jersey : World Scientific Publishing Company, 2017. — 860 p.
- Fung, Y.C., Tong, P., and Chen, X., Classical and Computational Solid Mechanics, New Jersey: World Scientific Publishing Company, 2017.
- Samarskii, A.A. The Theory of Difference Schemes / A.A Samarskii. — New York : Marcel Dekker, 2001. — 786 p.
- Samarskii, A.A., The Theory of Difference Schemes, New York: Marcel Dekker, 2001.
- LeVeque, R.J. Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems / R.J. LeVeque. — Philadelphia : SIAM, 2007. — 328 p.
- LeVeque, R.J., Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, Philadelphia: SIAM, 2007.
- Самарский, А.А. Устойчивость разностных схем / А.А. Самарский, А.В. Гулин. — М. : Наука, 1973. — 415 с.
- Samarskii, A.A. and Gulin, A.V., Ustojchivost’ raznostnyh skhem (Stability of Difference Schemes), Moscow: Nauka, 1973.
- Samarskii, А.А. Difference Schemes with Operator Factors / A.A Samarskii, P.P. Matus, P.N. Vabishchevich. — Dordrecht : Kluwer Academic Publishers, 2002. — 384 p.
- Samarskii, A.A., Matus, P.P., and Vabishchevich, P.N., Difference Schemes with Operator Factors, Dordrecht: Kluwer Academic Publishers, 2002.
- Ascher, U.M. Implicit-explicit methods for time-dependent partial differential equations / U.M. Ascher, S.J. Ruuth, B.T.R. Wetton // SIAM J. Numer. Anal. — 1995. — V. 32, № 3. — P. 797–823.
- Ascher, U.M., Ruuth, S.J., and Wetton, B.T.R., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 1995, vol. 32, no. 3, pp. 797–823.
- Hundsdorfer, W.H. Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Tquations / W.H. Hundsdorfer, J.G. Verwer. — Berlin : Springer, 2003. — 471 p.
- Hundsdorfer, W.H. and Verwer, J.G., Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations, Berlin: Springer, 2003.
- Marchuk, G.I. Splitting and alternating direction methods / G.I. Marchuk // Handbook of Numerical Analysis / Eds. P.G. Ciarlet and J.L. Lions. — North-Holland, 1990. — Vol. I. — P. 197–462.
- Marchuk, G.I., Splitting and alternating direction methods, Handbook of Numerical Analysis, Eds. P.G. Ciarlet and J.L. Lions, 1990, vol. I, pp. 197–462.
- Vabishchevich, P.N. Additive Operator-Difference Schemes: Splitting Schemes / P.N. Vabishchevich. — Berlin : De Gruyter, 2013. — 354 p.
- Vabishchevich, P.N., Additive Operator-Difference Schemes: Splitting Schemes, Berlin: De Gruyter, 2013.
- Vabishchevich P.N. Operator-difference scheme with a factorized operator / P.N. Vabishchevich // Large-Scale Scientific Computing 10th Int. Conf. — Sozopol, Bulgaria, June 8–12, 2015. — P. 72–79.
- Vabishchevich, P.N., Operator-difference scheme with a factorized operator, Large-Scale Scientific Computing 10th Int. Conf., Sozopol, Bulgaria, June 8–12, 2015, pp. 72–79.
- Vabishchevich P.N. Splitting schemes for some second-order evolutionary equations / P.N. Vabishchevich // Int. J. Numer. Anal. Model. — 2022. — V. 19, № 1. — P. 19–32.
- Vabishchevich P.N., Splitting schemes for some second-order evolutionary equations, Int. J. Numer. Anal. Model., 2022, vol. 19, no. 1, pp. 19–32.
Supplementary files
