Stability investigation for a certain explicit difference scheme (Q5899995)

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.