Discrete boundary treatment for the shifted wave equation in second-order form and related problems

We present strongly stable semi-discrete finite difference approximations to the quarter space problem (x > 0, t > 0) for the first order in time, second order in space wave equation with a shift term. We consider space-like (pure outflow) and time-like boundaries, with either second- or fourth-order accuracy. These discrete boundary conditions suggest a general prescription for boundary conditions in finite difference codes approximating first order in time, second order in space hyperbolic problems, such as those that appear in numerical relativity. As an example we construct boundary conditions for the Nagy–Ortiz–Reula formulation of the Einstein equations coupled to a scalar field in spherical symmetry.