Well-posedness of one-way wave equations and absorbing boundary conditions

Abstract

A one-way wave equation is a partial differential equation which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. Such equations are used in geophysics, in underwater acoustics, and as numerical "absorbing boundary conditions". Their construction can be reduced to the approximation of 1 − s 2 \sqrt {1 - {s^2}} on [ − 1 , 1 ] [ - 1,1] by a rational function r ( s ) = p m ( s ) / q n ( s ) r(s) = {p_m}(s)/{q_n}(s) . This paper characterizes those rational functions r for which the corresponding one-way wave equation is well posed, both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if r ( s ) r(s) interpolates 1 − s 2 \sqrt {1 - {s^2}} at sufficiently many points in ( − 1 , 1 ) ( - 1,1) , then well-posedness is assured. It follows that absorbing boundary conditions based on Padé approximation are well posed if and only if (m, n) lies in one of two distinct diagonals in the Padé table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived by Chebyshev or least-squares approximation.