Stability of finite difference schemes approximation for hyperbolic boundary value problems in an interval

Abstract:

In this article we are interested in the stability of finite difference schemes approximation for hyperbolic boundary value problems defined on the interval $\left [0,1 \right ]$. The seminal work of Gustafsson, Kreiss, and Sundström [Math. Comp. 26 (1972), pp. 649-686], mainly devoted to the half-line, gives a necessary and sufficient invertibility condition ensuring the stability of the scheme, the so-called discrete uniform Kreiss-Lopatinskii condition. An interesting point is that this condition is a discretized version of the one imposed in the continuous setting to ensure the strong well-posedness of the hyperbolic boundary value problem. However as pointed by Gustafsson, Kreiss, and Sundström [Math. Comp. 26 (1972), pp. 649-686] and as soon as several boundary conditions are concerned the solution to the scheme may develop an exponential growth with respect to the discrete time variable. The question addressed here is to characterize the schemes having this growth or not. This is made under a new invertibility condition which is a discretized version of the ones preventing the exponential growth in time of the solution to continuous hyperbolic boundary value problems in the strip studied by Benoit [Indiana Univ. Math. J. 69 (2020), pp. 2267-2323]. In some sense it shows that this continuous to discrete extension of the characterization occurs in the interval like in the half-line.