Complete Radiation Boundary Conditions: Minimizing the Long Time Error Growth of Local Methods

Abstract

We construct and analyze new local radiation boundary condition sequences for first-order, isotropic, hyperbolic systems. The new conditions are based on representations of solutions of the scalar wave equation in terms of modes which both propagate and decay. Employing radiation boundary conditions which are exact on discretizations of the complete wave expansions essentially eliminates the long time nonuniformities encountered when using the standard local methods (perfectly matched layers or Higdon sequences). Specifically, we prove that the cost in terms of degrees-of-freedom per boundary point scales with $\ln\frac{1}{\epsilon}\cdot\ln\frac{cT}{\delta}$, where $\epsilon$ is the error tolerance, T is the simulation time, and $\delta$ is the separation between the source-containing region and the artificial boundary. Choosing $\delta\sim\lambda$, where $\lambda$ is the wavelength, leads to the same estimate which has been obtained for optimal nonlocal approximations. Numerical experiments confirm that the efficiencies predicted by the theory are attained in practice.