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Stability investigation for a certain explicit difference scheme. (English. Russian original) Zbl 0711.65075

Sib. Math. J. 31, No. 1, 26-30 (1990); translation from Sib. Mat. Zh. 31, No. 1(179), 34-38 (1990).
The mixed problem for the symmetric t-hyperbolic system with real constant matrix coefficients \((1)\quad A\vec U_ t+B\vec U_ x+C\vec U_ y=\vec O,\quad \vec U^ I=S\vec U^{II}\quad at\quad x=0,\vec U(0,x,y)=\vec U_ 0(x,y)\) is considered. A, B, C are square \(N\times N\) matrices, A, B diagonal and A has all positive diagonal elements, \(\vec U=(\vec U^ I,\vec U^{II},\vec U^{III})^ T\), \(B\vec U=(\vec U^ I,-\vec U^{II},\vec O)^ T\), S a rectangular matrix. It is suggested that the boundary conditions are strictly dissipative \(-(B\vec U,\vec U)|_{x=0}\geq k_ 0(\vec U^{II},\vec U^{II})|_{x=0},\quad k_ 0>0.\) A two-level difference scheme for problem (1) is constructed using a discrete analogue of the energy integral, and its stability in energy norm is established.
Reviewer: V.L.Makarov

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
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[1] A. M. Blokhin, Energy Integrals and Their Applications in Problems of Gas Dynamics [in Russian], Nauka, Novosibirsk (1986).
[2] A. M. Blokhin and R. D. Alaev, ?Stability of a modified MacCormack difference scheme for a symmetric t-hyperbolic system,? in: Well-Posed Boundary Value Problems for Nonclassical Equations of Mathematical Physics [in Russian], Akad. Nauk SSSR, Sib. Otd., Inst. Mat., Novosibirsk (1984), pp. 24-42. · Zbl 0582.65075
[3] A. M. Blokhin, R. D. Alaev, and I. Yu. Druzhinin, ?The stability of explicit difference schemes for symmetric t-hyperbolic systems,? in: Boundary Value Problems for Partial Differential Equations [in Russian], Novosibirsk Gos. Univ., Inst. Mat., Novosibirsk (1984), pp. 26-39.
[4] S. K. Godunov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1979). · Zbl 0447.22011
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