Boundary Gauge Wave and Shifted Gauge Wave with Amp=0.5 Test Results for the ABIGEL Code

For the case without periodic boundaries in the $x$-direction, the boundary data are provided at $x=\pm .5$. For example, in the 1D simulation, for a formulation with a Sommerfeld boundary condition on the metric component $g_{xx}$ the correct boundary data are

$$(\partial_t+\partial_x)g_{xx}\vert _{x=.5} = 0$$ (14)

$$(\partial_t-\partial_x)g_{xx}\vert _{x=-.5} = 2\partial_t H(-.5-t).$$ (15)

With this inhomogeneous Sommerfeld data, the wave enters through the boundary at $x=-.5$, propagates across the grid and exits through the boundary at $x=.5$.