Hybrid Spectral Difference Methods for Elliptic Equations on Exterior Domains with the Discrete Radial Absorbing Boundary Condition

Abstract

The hybrid spectral difference methods (HSD) for the Laplace and Helmholtz equations in exterior domains are proposed. We consider the fictitious domain method with the absorbing boundary conditions (ABCs). The HSD 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 fictitious domain is composed of two subregions; the Cartesian grid region and the boundary layer region in which the radial grid is imposed. The boundary layer region with the radial grid makes it easy to implement the discrete radial ABC. The discrete radial ABC is a discrete version of the Bayliss-GunzburgerTurkel ABC without pertaining any radial derivatives. Numerical experiments confirming efficiency of our numerical scheme are provided.

Effects of concentrated elements on vibro-acoustic characteristics of a circular ring considering both external and internal fluids

“…Among them, finite element method (FEM) (Brenner et al, 2018) and boundary element method (BEM) (Venas and Kvamsdal, 2020) are the two commonly used methods. The BEM inherently satisfies the Sommerfeld radiation condition for the exterior acoustic problems, whereas some special mathematical treatments need to be incorporated with the FEM to model the infinite exterior acoustic domain, such as perfectly matched layer (PML) (Bunting et al, 2018), infinite elements (Wu and Xiang, 2019), absorbing boundary condition (Jeon, 2018), and Dirichlet to Neumann mapping (Kapita and Monk, 2018). However, the system matrices yielded in the FEM are sparse and symmetric, which can significantly reduce the computation cost and storage requirement.…”

Immersed hybrid difference methods for elliptic boundary value problems by artificial interface conditions

“…The hybrid difference (HD) method is a finite difference version of the hybridized discontinuous Galerkin method and it was introduced by the authors and their colleagues for the elliptic, Stokes and Navier-Stokes equations [15,16,17,18]. Recently, the immersed boundary approach was applied to the HD method to develop the IHD method by the first author [14] for elliptic interface problems.…”