TY - JOUR
T1 - Direct implementation of high order BGT artificial boundary conditions
AU - Medvinsky, M.
AU - Tsynkov, S.
AU - Turkel, E.
N1 - Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2019/1/1
Y1 - 2019/1/1
N2 - Local artificial boundary conditions (ABCs) for the numerical simulation of waves have been successfully used for decades (most notably, the boundary conditions due to Engquist & Majda, Bayliss, Gunzburger & Turkel, and Higdon). The basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution. The larger the number of terms canceled, the higher the order of the boundary condition and, in turn, the smaller the reflection error due to truncation of the original unbounded domain by an artificial outer boundary. In practice, however, the use of local ABCs has been limited to low orders (first and second), because higher order boundary conditions involve higher order derivatives of the solution, which may harm well-posedness and cause numerical instabilities. They are also difficult to implement especially in finite elements. A prominent exception is the development of local high order ABCs based on auxiliary variables. In the current paper, we implement high order Bayliss–Turkel ABCs directly — with no auxiliary variables yet no discrete approximation of the constituent high order derivatives either. Instead, we represent the solution at the boundary as an expansion with respect to a continuous basis. For the spherical artificial boundary, the basis consists of eigenfunctions of the Beltrami operator (spherical harmonics), which enable replacing the high order derivatives in the ABCs with powers of the corresponding eigenvalues. The continuous representation at the boundary is coupled to higher order compact finite differences inside the domain by the method of difference potentials (MDP). It maintains high order accuracy even when the boundary is not aligned with the discretization grid.
AB - Local artificial boundary conditions (ABCs) for the numerical simulation of waves have been successfully used for decades (most notably, the boundary conditions due to Engquist & Majda, Bayliss, Gunzburger & Turkel, and Higdon). The basic idea behind these boundary conditions is that they cancel several leading terms in an expansion of the solution. The larger the number of terms canceled, the higher the order of the boundary condition and, in turn, the smaller the reflection error due to truncation of the original unbounded domain by an artificial outer boundary. In practice, however, the use of local ABCs has been limited to low orders (first and second), because higher order boundary conditions involve higher order derivatives of the solution, which may harm well-posedness and cause numerical instabilities. They are also difficult to implement especially in finite elements. A prominent exception is the development of local high order ABCs based on auxiliary variables. In the current paper, we implement high order Bayliss–Turkel ABCs directly — with no auxiliary variables yet no discrete approximation of the constituent high order derivatives either. Instead, we represent the solution at the boundary as an expansion with respect to a continuous basis. For the spherical artificial boundary, the basis consists of eigenfunctions of the Beltrami operator (spherical harmonics), which enable replacing the high order derivatives in the ABCs with powers of the corresponding eigenvalues. The continuous representation at the boundary is coupled to higher order compact finite differences inside the domain by the method of difference potentials (MDP). It maintains high order accuracy even when the boundary is not aligned with the discretization grid.
KW - Beltrami operator
KW - Expansion with respect to continuous basis
KW - Helmholtz equation
KW - Method of difference potentials (MDP)
KW - Propagation of waves over unbounded regions
KW - Spherical artificial boundary
UR - http://www.scopus.com/inward/record.url?scp=85054066683&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2018.09.040
DO - 10.1016/j.jcp.2018.09.040
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AN - SCOPUS:85054066683
SN - 0021-9991
VL - 376
SP - 98
EP - 128
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -