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Article information
2000 , Volume 5, ¹ 4, p.82-96
Nemirovskii Y.V., Yankovskii A.P.
Numerical integration of two-dimensional boundary problems with large solution gradients
The idea of Runge-Kutta methods generalization to the two-dimensional case of approximated integration of initial-boundary problems corresponding to the partial differential equations has been put forward and tested. It is shown that some classical finite-difference schemes of integrating the equations of transfer and non-stationary unidimensional heat conductivity can be obtained as a corollary of such a generalization. New schemes with a high order of accuracy have been obtained for various problems of mathematical physics. The stability of these schemes has been proved and the calculation results for the problems with large solution gradients have been presented. There are given specific examples, which demonstrate that the classical lower-order schemes inadequately describe the solution of such problems, whereas the high-order schemes constructed by means of the suggested generalized Runge-Kutta methods yield good approximations to the accurate solutions.
[full text] Classificator Msc2000:- *35K05 Heat equation
- 35K15 Initial value problems for second-order, parabolic equations
- 65L06 Multistep, Runge-Kutta and extrapolation methods
- 65M06 Finite difference methods
- 65M12 Stability and convergence of numerical methods
- 74E30 Composite and mixture properties
Keywords: two-dimensional boundary value problem, transport equation, generalized Runge-Kutta method, stability and convergence, heat equation, composite materials, finite difference schems
Author(s):
Nemirovskii Yurii Vladimirovich
Office: Institute of Theoretical and Applied Mechanics of SB RAS
Address: 630090, Russia, Novosibirsk, 4/1 Institutskaya str.
Phone Office: 354273
E-mail: fomin@icans.nsk.su
Yankovskii A P
Office: Institute of Theoretical and Applied Mechanics of SB RAS
Address: 630090, Russia, Novosibirsk, 4/1 Institutskaya str.
Bibliography link:
Nemirovskii Y.V., Yankovskii A.P. Numerical integration of two-dimensional boundary problems with large solution gradients // Computational technologies. 2000. V. 5. ¹ 4. P. 82-96
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