Mathematics and Statistics 7(3): 82-89, 2019 DOI: 10.13189/ms.2019.070305 http://www.hrpub.org The Difference Splitting Scheme for Hyperbolic Systems with Variable Coefficients Aloev R. D. 1,∗ , Eshkuvatov Z. K. 2 , Khudoyberganov M. U. 1 , Nematova D. E. 1 1 Faculty of Mathematics, National University of Uzbekistan (NUUz), Tashkent, Uzbekistan 2 Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), Negeri Sembilan, Malaysia Copyright c 2019 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract In the paper, we propose a systematic approach to design and investigate the adequacy of the computational models for a mixed dissipative boundary value problem posed for the symmetric t-hyperbolic systems. We consider a two-dimensional linear hyperbolic system with variable coef- ficients and with the lower order term in dissipative boundary conditions. We construct the difference splitting scheme for the numerical calculation of stable solutions for this system. A discrete analogue of the Lyapunov’s function is constructed for the numerical verification of s tability of s olutions for the considered problem. A priori estimate is obtained for the discrete analogue of the Lyapunov’s function. This estimate allows us to assert the exponential stability of the numerical solution. A theorem on the exponential stability of the solution of the boundary value problem for linear hyperbolic system and on stability of difference splitting scheme in the Sobolev spaces was proved. These stability theorems give us the opportunity to prove the convergence of the numerical solution. Keywords Difference Scheme, Lyapunov Function, Mixed Problem, Stability 1 Introduction We consider the mixed dissipative boundary value problem for a two-dimensional linear hyperbolic system with variable co- efficients and lower order term [1]. For this problem, we con- struct and investigate the difference splitting scheme in order to obtain stable solutions. A discrete analogue of the Lyapunov function is constructed and an a-priori estimate is obtained for it. The obtained a-priori estimate allows us to assert the expo- nential stability of the numerical solution. It should be noted that numerous problems have been de- voted to the solution of such problems (see [3]-[10] and ref- erences in them). Studying the stability of solutions for one- dimensional hyperbolic systems is the subject of [2]. Authors of the paper investigate the stability of the solution by con- structing the Lyapunov function and using a priori estimates for solution in various functional spaces. However, stability of the difference schemes, constructed in all these papers, was investigated using the technique of constructing dissipative en- ergy integrals. The a-priori estimates are obtained in these pa- pers. But they are not enough to get exponential stability of the numerical solution. 2 Differential statement of the problem In the domain G = {(t, x, y):0 <t ≤ T, 0 < x < l, −∞ < y< +∞}, we considered a symmetric hyperbolic system in a special canonical form ∂ v ∂t + K ∂ v ∂x + C ∂ v ∂y + Mv = 0 (1) with boundary conditions for x =0: v I = sv II (2) for x = l : v II = rv I (3) and with initial data at t =0 v i (0, x, y)= ϕ i (x, y),i =1,...,n, 0 ≤ x ≤ l, −∞ ≤ y ≤ +∞ (4) where v I = (v 1 ,v 2 ,...,v m ) T , v II = (v m+1 ,v m+2 ,...,v n ) T , K is a diagonal matrix, C is a positive definite matrix, M is a n-th order square real matrix v =  v I v II  , K =  K + 0 0 −K −  , K + =      k 1 0 ··· 0 0 k 2 ··· 0 . . . . . . . . . . . . 0 ··· 0 k m      , K − =      k m+1 0 ··· 0 0 k m+2 ··· 0 . . . . . . . . . . . . 0 ··· 0 k n      ,k i > 0,i =1,...,n;