Application of a complete radiation boundary condition for the Helmholtz equation in locally perturbed waveguides

This paper deals with an application of a complete radiation boundary condition (CRBC) for the Helmholtz equation in locally perturbed waveguides. The CRBC, one of efficient high-order absorbing boundary conditions, has been analyzed in straight waveguides in Hagstrom and Kim (2019). In this paper, we apply CRBC to the Helmholtz equation posed in locally perturbed waveguides and establish the well-posedness of the problem and convergence of CRBC approximate solutions. The new CRBC proposed in this paper improves the one studied in Hagstrom and Kim (2019) in two aspects. The first one is that the new CRBC involves more damping parameters with the same computational cost as that of CRBC in Hagstrom and Kim (2019), which results in 50% smaller reflection errors. The second one is that the new CRBC takes a Neumann terminal condition of three term recurrence relations of auxiliary variables instead of a Dirichlet terminal condition used in Hagstrom and Kim (2019) so that it can treat cutoff modes effectively. Finally, we present numerical experiments illustrating the convergence theory.