TY - JOUR
T1 - Sixth-Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect
AU - Feng , Qiwei
AU - Han , Bin
AU - Michelle , Michelle
JO - Communications in Computational Physics
VL - 3
SP - 672
EP - 712
PY - 2023
DA - 2023/10
SN - 34
DO - http://doi.org/10.4208/cicp.OA-2023-0062
UR - https://global-sci.org/intro/article_detail/cicp/22021.html
KW - Helmholtz equation, finite difference, pollution effect, interface, pollution minimization, mixed boundary conditions, corner treatment.
AB - <p style="text-align: justify;">Due to its highly oscillating solution, the Helmholtz equation is numerically
challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller
than the reciprocal of the wavenumber is typically required (known as the pollution
effect). High-order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high-order compact finite difference method
for 2D Helmholtz equations with singular sources, which can also handle any possible
combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method is sixth-order consistent for a constant wavenumber, and
fifth-order consistent for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average
truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several
state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $kh$ is near 1 with $k$ being the wavenumber and $h$ the mesh size.</p>