A general approach for high order absorbing boundary conditions for the Helmholtz equation

Abstract

When solving a scattering problem in an unbounded space, one needs to implement the Sommerfeld condition as a boundary condition at infinity, to ensure no energy penetrates the system. In practice, solving a scattering problem involves truncating the region and implementing a boundary condition on an artificial outer boundary. Bayliss, Gunzburger and Turkel (BGT) suggested an Absorbing Boundary Condition (ABC) as a sequence of operators aimed at annihilating elements from the solution’s series representation. Their method was practical only up to a second order condition. Later, Hagstrom and Hariharan (HH) suggested a method which used auxiliary functions and enabled implementation of higher order conditions.We compare various absorbing boundary conditions (ABCs) and introduce a new method to construct high order ABCs, generalizing the HH method. We then derive from this general method ABCs based on different series representations of the solution to the Helmholtz equation – in polar, elliptical and spherical coordinates. Some of these ABCs are generalizations of previously constructed ABCs and some are new.These new ABCs produce accurate solutions to the Helmholtz equation, which are much less dependent on the various parameters of the problem, such as the value of k, or the eccentricity of the ellipse. In addition to constructing new ABCs, our general method sheds light on the connection between various ABCs. Computations are presented to verify the high accuracy of these new ABCs.