On convergence of difference schemes for two-dimensional second-order hyperbolic equations with discontinuous coefficients

V.A.Dobrushkin; V.A.Tsurko; G.M.Zayats

Three-layer symmetric difference schemes are proposed for solving two-dimensional hyperbolic differential equations with discontinuous variable coefficients. The coefficients in the problem are assumed to have finite discontinuities along the line which is perpendicular to the direction of a spatial variable. The convergence rate for difference approximations is derived using energy inequalities. The results of numerical experiments are presented.

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3. V.A.Dobrushkin, The boundary-value problems in the dynamic theory of elasticity for the wedge-shaped domains, Nauka i Tehnika, 1988. (in Russian)

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9. A.A.Samarskii, I.V.Fryazinov, On convergence of the homogeneous difference schemes for heat equation with discontinuous coefficients, Zh. Vychislit. Mat. Mat. Fiz., 1 (1961), № 5, 806--824. (in Russian)

10. I.V.Fryazinov, Economic finite difference schemes for multi dimensional heat transfer equation with discontinuous coefficients, Comput. Maths Math. Phys., 13 (1973), № 1, 80--91.

11. V.A.Tsurko, Finite-difference methods for approximate solutions of two- dimensional parabolic equations with discontinuous coefficients, Differential Equations, 38 (2002), № 7, 1070--1073.

12. A.A.Samarskii and E.S. Nikolaev, Methods for Solution of Mesh Equations, Moscow, Nauka, 1978. (in Russian)

	On convergence of difference schemes for two-dimensional second-order hyperbolic equations with discontinuous coefficients
Author	V.A.Dobrushkin, V.A.Tsurko, G.M.Zayats dobrush@dam.brown.edu, vtsurko@im.bas-net.by, zayats@im.bas-net.by Division of Applied Mathematics, Brown University,182 George St. Providence, RI 02912, USA, Institute of Mathematics, National Academy of Sciences,11 Surganova St., Minsk, 220072, Belarus
Abstract	Three-layer symmetric difference schemes are proposed for solving two-dimensional hyperbolic differential equations with discontinuous variable coefficients. The coefficients in the problem are assumed to have finite discontinuities along the line which is perpendicular to the direction of a spatial variable. The convergence rate for difference approximations is derived using energy inequalities. The results of numerical experiments are presented.
Keywords	convergence, difference scheme, two-dimensional second-order hyperbolic equations, discontinuous coefficient
DOI	doi:10.30970/ms.31.1.91-101
Reference	1. A.Meister and J.Struckmeier, Hyperbolic Partial Differential Equations, Vieweg, 2002. 2. R.J.Clifton, A difference method for plane problems in dynamic elasticity, Quarterly of Applied Mathematics, 25 (1967), № 1, 97--116. 3. V.A.Dobrushkin, The boundary-value problems in the dynamic theory of elasticity for the wedge-shaped domains, Nauka i Tehnika, 1988. (in Russian) 4. K.R.Kelly, R.W.Ward, S. Treitel, and R.M.Alford, Synthetic seismograms; A finite-difference approach, Geophysics, 41 (1976), № 1, 2--27. 5. R.D.Richtmyer, K.W.Morton, Difference methods for initial-value problems, New York-Sydney, 1967. 6. G.E.Forsythe, W.R.Wasow, Finite-difference methods for partial differential equations, John Wiley $\&$ Sons, inc. New York-London, 1961. 7. S.K.Godunov, V.S.Riaben'kii, Difference schemes, Moscow, Nauka, 1977. (in Russian) 8. A.A.Samarskii, The theory of difference schemes, Moscow, Nauka, 1977. (in Russian) 9. A.A.Samarskii, I.V.Fryazinov, On convergence of the homogeneous difference schemes for heat equation with discontinuous coefficients, Zh. Vychislit. Mat. Mat. Fiz., 1 (1961), № 5, 806--824. (in Russian) 10. I.V.Fryazinov, Economic finite difference schemes for multi dimensional heat transfer equation with discontinuous coefficients, Comput. Maths Math. Phys., 13 (1973), № 1, 80--91. 11. V.A.Tsurko, Finite-difference methods for approximate solutions of two- dimensional parabolic equations with discontinuous coefficients, Differential Equations, 38 (2002), № 7, 1070--1073. 12. A.A.Samarskii and E.S. Nikolaev, Methods for Solution of Mesh Equations, Moscow, Nauka, 1978. (in Russian)
Pages	91-101
Volume	31
Issue	1
Year	2009
Journal	Matematychni Studii
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