This interesting paper is concerned with stability of finite-difference methods for a hyperbolic system (1) \(u_ t=Au_ x\), where u(x,t) is N- vector, \(t>0\), x is a real number, A is a constant \(N\times N\) matrix. The problem (1) is approximated using the following \(s+2\)-level finite- difference formula \((2)\quad Q_{-1}v^{n+1}=\sum^{S}_{\sigma =0}Q_{\sigma}v^{n-\sigma},\) where \(v^ n_ j=v(jh,nk)\), k is the time step, h is the space step and each \(Q_{\sigma}\) is a spatial finite-difference operator with constant matrix coefficient of dimension \(N\times N.\)
The wave solutions to (2) of the form \((3)\quad v^ n_ j=V\kappa^ jz^ n,\) where \(V\in {\mathbb{R}}^ N\), \(\kappa =e^{i\xi h}\), \(z=e^{i\omega k}\) are investigated. The numbers \(\xi\in {\mathbb{R}}\), \(\omega\in {\mathbb{C}}\) are related by a dispersion relation \(\omega =\omega (\xi)\), and the group velocity is defined by \(C=-d\omega /d\xi\). A solution of (3) with \(| \kappa | =| z| =1\) and \(C\leq 0\) (resp. \(C\geq 0)\) or with \(| z| \geq 1\), \(| \kappa | >1\) (resp. \(| z| \geq 1\), \(| \kappa | <1)\) is called leftgoing (resp. rightgoing). The general solution to (2) is a linear combination of rightgoing and leftgoing waves with coefficients \(\alpha_ m\) and \(\beta_ m\), respectively. If the vectors \(\alpha\) and \(\beta\) are related by \(\alpha =B(z)\beta\), then B(z) is the so-called reflection matrix.
The author considers various connections between GKS stability introduced by B. Gustafsson, H.-O. Kreiss, and A. Sundström [ibid. 26, 649-686 (1972; Zbl 0293.65076)] and the so-called P-stability (practical stability). Unfortunately, both are too complicated to reproduce here. The numerical schemes considered contain two boundaries or two or more interfaces. It is a source of appearance of some additional “parasitic” waves which may cause instability. The author obtains several interesting results. Here only two of them are quoted: the first one says that for the stability of some multi-interface model it is not sufficient that the individual interfaces be stable. The second one establishes stability in certain two-interface problem if each interface individually is stable and the reflection matrices have norms less than one. The explicit examples of numerical schemes play an important role in the paper. Some of them are rather artificial but still well working.