We present a comparison of various methods of local absorbing boundary conditions (ABC) for numerical solutions of the Helmholtz equation exterior to an ellipse. We also introduce a new boundary condition based on a modal expansion in Mathieu functions. We compare this new ABC for an outer ellipse with the other boundary conditions.

Elliptic equations in exterior regions usually require a boundary condition at infinity to ensure the well-posedness of the problem. One of the practical applications is the Helmholtz equation. For computational reasons one needs to truncate the unbounded domain. Typically, truncation is implemented by the introduction of an artificial outer domain, together with absorbing boundary conditions along this domain, that reduce wave reflections into the physical domain. Bayliss with Turkel [3] and later with Gunzburger [2] presented a sequence of boundary conditions in polar and spherical coordinates. These boundary conditions were generalized by several authors when the artificial surface is not a circle or sphere.