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 coefficients 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 stability of solutions 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.
Highlights
We consider the mixed dissipative boundary value problem for a two-dimensional linear hyperbolic system with variable coefficients and lower order term [1]
The obtained a-priori estimate allows us to assert the exponential stability of the numerical solution
For the numerical solution of the mixed problem (1)-(4), we suggest a difference splitting scheme wI wII
Summary
We consider the mixed dissipative boundary value problem for a two-dimensional linear hyperbolic system with variable coefficients and lower order term [1] For this problem, we construct and investigate the difference splitting scheme in order to obtain stable solutions. Authors of the paper investigate the stability of the solution by constructing the Lyapunov function and using a priori estimates vi(0, x, y) = φi(x, y), i = 1, . The system (1) with boundary conditions (2)-(3) is said to be exponentially stable in the L2− norm if there exists such ν > 0 and c > 0 that for any initial condition φ ∈ L2 ((0, l), (−∞, +∞), Rn) , the L2solution of the mixed problem (1)-(4) satisfies the inequality v(t, ·) L2((0,l),(−∞,+∞);Rn) ≤. (v, v) dxdy we obtain v(t, ·) L2((0,l),(−∞,+∞),Rn) ≤ ≤ γe−νt/2 Φ L2((0,l),(−∞,+∞),Rn) , t ∈ [0, +∞)
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