On the stability of the high-order Higdon Absorbing Boundary Conditions

The Higdon sequence of Absorbing Boundary Conditions (ABCs) for the linear wave equation is considered. Building on a previous work of Ha-Duong and Joly, which related to other forms of boundary conditions, the Higdon ABCs are proved to be energy-stable (on the continuous level) up to any order. This type of stability is stronger than the more standard notion of stability of initial boundary value problems in the sense of Kreiss; in particular it leads to stability estimates which are uniform in time. In consequence to the theorem proved here, energy-stability is immediately implied for the high-order Givoli-Neta and Hagstrom-Warburton ABCs, which are reformulations of the Higdon ABCs using auxiliary variables. A weakness of this theory is that it requires sufficiently smooth data, and that the required smoothness level increases with the order of the ABC. This issue and its implications are discussed.