UDC 517:519.6
1 V.M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine
gladky@ukr.net
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ABOUT ONE SPLITTING SCHEME FOR DIFFUSION AND HEAT CONDUCTION PROBLEMS
Abstract. The problem of mathematical modeling and optimization of diffusion and heat conduction non-stationary processes is considered. An approach that uses the idea of splitting and computation of the obtained difference schemes using explicit schemes of running counting is proposed for the numerical solution of multidimensional diffusion and heat conduction initial-boundary-value problems. Construction of difference splitting schemes and approximation and stability on the initial data are investigated. Differential properties of the quality functional are analyzed for the numerical solution of the optimal control problem for a parabolic equation. An iterative algorithm for determining the optimal control is proposed.
Keywords: parabolic equation, optimal control problem, numerical method, splitting methods, difference scheme, stability.
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REFERENCES
- Marchuk G.I. Mathematical modeling in the environmental problem [in Russian]. Moscow: Nauka, 1982. 320 p.
- Egorov A.I. Optimal control of thermal and diffusion processes [in Russian]. Moscow: Nauka, 1978. 463 p.
- Egorov A.I. Fundamentals of management theory [in Russian]. Moscow: Fizmatgiz, 2004. 504 p.
- Zgurovsky M.Z., Skopetskiy V.V., Khrushch V.K., Belyaev N.N. Numerical modeling of the spread of pollution in the environment [in Russian]. Kiev: Nauk. dumka. 1997. 368 p.
- Sergienko I.V., Skopetskii V.V., Deineka V.S. Mathematical modeling and research of processes in heterogeneous environments [in Russian]. Kiev: Nauk. dumka, 1991. 432 p.
- Samarskii A.A., Vabishchevich P.N. Computational heat transfer [in Russian]. Moscow: Editorial URSS, 2003. 784 p.
- Sauliev V.K. Integration of parabolic equations using the grid method [in Russian]. Moscow: Fizmatgiz, 1960. 324 p.
- Samarskii A.A., Vabishchevich P.N. Numerical methods for solving inverse problems of mathematical physics. Berlin: Walter de Gruyter, 2007. 438 p.
- Vabishchevich P.N., Vasil’ev V.I. Computational algorithms for solving the coefficient inverse problem for parabolic equations. Inverse Probl. Sci. Engin. 2016. Vol. 24, N. 1. P. 42–59.
- Vabishchevich P.N., Vasilieva M.V., Vasiliev V.I. Computational identification of the right side of a parabolic equation. Zhurn. vychisl. matem. i matem. fiziki, 2015. Vol. 55, N 6. P. 1020–1027.
- Godunov S.K., Ryaben'ky V.S. Difference schemes [in Russian]. Moscow: Nauka, 1977. 440 p.
- Agoshkov V.I. Optimal control methods and conjugate equations in problems of mathematical physics [in Russian]. Moscow: INM RAS, 2004. 256 p.
- Deineka V.S., Sergienko I.V. Models and methods for solving problems in heterogeneous environments [in Russian]. Kiev: Nauk. dumka, 2001. 606 p.
- Ilyin V.P. Finite difference and finite volume methods for elliptic equations [in Russian]. Novosibirsk: Publishing House IM SB RAS, 2001. 318 p.
- Vabishchevich P.N., Zakharov P.E. Explicit-implicit splitting schemes for parabolic equations and systems. Numerical methods and applications. Springer, 2015. P. 157–166.
- Marchuk G.I. Cleavage methods [in Russian]. Moscow: Nauka, 1988. 264 p.
- Gladky A.V. Stability of difference splitting schemes for convection diffusion equation. Cybernetics and Systems Analysis. 2017. Vol. 53, N 2. P. 193–203.
- Gladky A.V. Analysis of splitting algorithms in convection–diffusion problems. Cybernetics and Systems Analysis. 2014. Vol. 50, N 4. P. 548–559.
- Vabishchevich P.N. On a new class of additive (splitting) operator sub difference schemes. Math. Comput. 2012. Vol. 81, N 277. P. 267–276.
- Vabishchevich P.N. Flux-splitting schemes for parabolic problems. Computational Mathematics and Mathematical Physics. 2012. Vol. 52, N 8. P. 1128–113.
- Samarsky A.A., Gulin A.V. Stability of difference schemes [in Russian]. Moscow: Nauka, 1973. 416 p.
- Marchuk G.I., Agoshkov V.N. Introduction to projection-grid methods [in Russian]. Moscow: Nauka, 1981. 416 p.
- Alifanov O.M. Extreme methods for solving incorrect tasks [in Russian]. Moscow: Nauka, 1988. 288 p.
- Vasiliev P.F. Optimization methods [in Russian]. Moscow: Factorial Press, 2002. 824 p.