Application of the Complete Radiation Boundary Condition and the Rational Absorbing Boundary Condition in $\mathbb{R}^2$ and $\mathbb{R}^3$

Abstract

In this study, we explore two distinct rational approximations to the radiation condition for effectively solving time-harmonic wave propagation problems governed by the Helmholtz equation in $\mathbb{R}^d,$ $d = 2$ or $3.$ First, we focus on the well-known complete radiation boundary condition (CRBC), which was developed for a transparent boundary condition for two-dimensional problems. The extension of CRBC to three-dimensional problems is a primary concern. Applications of CRBC require removing a near-cutoff region for a frequency range of a process to minimize reflection errors. To address the limitation faced by the CRBC application we introduce another absorbing boundary condition that avoids this demanding truncation. It is a new rational approximation to the radiation condition, which we call a rational absorbing boundary condition, that is capable of accommodating all types of propagating wave modes, including the grazing modes. This paper presents a comparative performance assessment of two approaches in two and three-dimensional spaces, providing insights into their effectiveness for practical application in wave propagation problems.