Cauchy boundaries in linearized gravitational theory

We investigate the numerical stability of Cauchy evolution of linearized gravitational theory in a three-dimensional bounded domain. Criteria of robust stability are proposed, developed into a testbed and used to study various evolution-boundary algorithms. We construct a standard explicit finite difference code which solves the unconstrained linearized Einstein equations in the $3+1$ formulation and measure its stability properties under Dirichlet, Neumann, and Sommerfeld boundary conditions. We demonstrate the robust stability of a specific evolution-boundary algorithm under random constraint violating initial data and random boundary data.