High-Order Non-Reflecting Boundary Scheme for Time-Dependent Waves

Abstract

A new non-reflecting boundary scheme is proposed for time-dependent wave problems in unbounded domains. The linear time-dependent wave equation, with or without a disper- sive term, is considered outside of an obstacle or in a semi-infinite wave guide. The infinite domain is truncated via an artificial boundary B, and a high-order Non-Reflecting Boundary Condition (NRBC) is imposed on B. Then the problem is solved numerically in the finite domain bounded by B. The new boundary scheme is based on a reformulation of the se- quence of NRBCs proposed by Higdon. In contrast to the original formulation of the Higdon conditions, the scheme constructed here does not involve any high derivatives beyond second order. This is made possible by introducing special auxiliary variables on B. As a result, the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. The param- eters appearing in the NRBC are chosen automatically via a special procedure. Numerical examples concerning a semi-infinite wave guide are used to demonstrate the performance of the new method.