High order numerical sommerfeld boundary conditions: Theory and experiments

Abstract

We develop high-order, non-reflecting boundary equations for a semidiscrete approximation of the simple (hyperbolic) advection equation U 1 + cU x = 0. These boundary equation are based on a discrete interpretation of Sommerfeld's radiation condition for a second order wave equation which is associated with the semi-discrete equation. The performance of these schemes is expressed by an exact measure of the energy reflected at the boundary. For low order cases, the discrete Sommerfeld boundary equations are identical with the standard finite difference equations, but for high orders of approximation (starting with 4 points), the discrete Sommerfeld schemes differ from standard finite differences. It is shown, and verified experimentally, that the discrete Sommerfeld schemes are optimal, in the sense that they produce the least amount of reflected energy. Moreover, it is known theoretically, and we verify experimentally, that the reflected energy remains invariat when the semi-discrete equation are time-discretized with the trapezoidal (Crank-Nicolson) method. The corresponding fully discrete boundary equations are thus also optimal in the sense that they minimize the reflected energy.