The Crank–Nicolson hybrid difference method for the time wave equation on exterior domains with the discrete radial absorbing boundary condition

Abstract

The hybrid difference method (HDM) combined with the discrete radial absorbing boundary condition (ABC) and the Crank–Nicolson time marching is presented. The HDM method is a finite difference version of the hybridized Galerkin method, and it consists of two types of finite difference approximations; the cell finite difference and the interface finite difference. The discrete radial ABC is a time dependent version of the discrete radial ABC for the elliptic equations, that has been proposed in [Y. Jeon, Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition, J. Sci. Comput. 75, 889–905 (2018)]. Through a change of variables the Bayliss–Turkel ABC is renewed. Moreover, it is partially proved that the discrete radial ABC is a non-standard finite difference approximation of the Bayliss–Turkel ABC. The discrete radial ABC is easy to implement and can be applied to an arbitrary convex fictitious domain including the rectangular box since it is degenerated to a quasi-Dirichlet condition for the time dependent problem. Two-dimensional numerical experiments confirming efficiency of our numerical scheme for various wave numbers and several artificial boundaries are presented.