Let us look at some of the details of the reflection coefficient calculation. A unit amplitude, monochromatic plane wave incident on the side boundary generates a reflected wave of magnitude c. The mathematical representation is:
The case of zero reflection arises when the numerical value of kz selected by the interior equation at happens also to satisfy exactly the dispersion relation D of the side boundary condition. This explains why we try to match the quarter-circle as closely as possible. The straight-line dispersion relation does not correspond to the most general form of a side boundary condition, which is expressible on just two end points. A more general expression with adjustable parameters b1, b2, and b3, which fits even better, is
The absolute stability of straight-line absorbing side boundaries for the 15 equation can be established, including the discretization of the x-axis. Unfortunately, an airtight analysis of stability seems to be outside the framework of the Muir impedance rules. As a consequence, I don't believe that stability has been established for the 45 equation.