An Efficient Finite Difference-Based PML Technique for Acoustic Scattering Problems

TL;DR

The work tackles acoustic scattering in an unbounded domain by coupling perfectly matched layers with high-order compact finite difference methods in polar coordinates. A novel pollution-minimization technique directly optimizes stencil coefficients to reduce dispersion errors, enabling baseline fourth-order accuracy that, when augmented with exponential stretching and stencil enlargement, achieves sixth-order consistency. The method supports multiple, non-circular scatterers and remains robust across a broad range of wavenumbers and PML configurations, with numerical results showing dramatic reductions in error. This approach offers a computationally efficient, sparse, and accurate framework for large-scale Helmholtz scattering problems in complex geometries.

Abstract

The acoustic scattering problem is modeled by the exterior Helmholtz equation, which is challenging to solve due to both the unboundedness of the domain and the high dispersion error, known as the pollution effect. We develop high-order compact finite difference methods (FDMs) in polar coordinates to numerically solve the problem with multiple arbitrarily shaped scatterers. The unbounded domain is effectively truncated and compressed via perfectly matched layers (PMLs), while the pollution effect is handled by the high order of our method and a novel pollution minimization technique. This technique is easy to implement, rigorously proven to be effective and shows superior performance in our numerous numerical results. The FDMs we propose in regular polar coordinates achieve fourth consistency order. Yet, combined with exponential stretching and mesh refinement, we can reach sixth consistency order by slightly enlarging the stencil at certain locations. Our numerical examples demonstrate that the proposed FDMs are effective and robust under various wavenumbers, PML layer thickness and shapes of scatterers.

Figures (9)

• Figure 1: Comparison of meshes generated in $(r, \theta)$ (left) and $(s, \theta)$ coordinates (right). The mesh size at the interface is the same. The bottom figures are zoomed-in views of the corresponding top figures at the origin. Red circle indicates the interface $\Gamma$.
• Figure 2: Result of mesh refinement under the exponentially stretched mesh. In the second figure, the dangling nodes and auxiliary nodes for the FDM are marked in empty squares and crosses, respectively.
• Figure 2: Error and convergence order using three different meshes in \ref{['ex:ex1']}. The mesh size $h$ at the interface is the same for all data on the same row. The order corresponding to mesh size $h$ is calculated by $\log (e_{h'} / e_h) / \log (h' / h)$, where $h'$ is the previous mesh size.
• Figure 3: Different types of stencils with stencil center marked in blue and other stencil points marked in black. Stencils centered at an auxiliary node in $\Omega$ are not shown. The red curve, empty squares and crosses are the same as in \ref{['fig:refinement']}.
• Figure 4: Real part of reference solution $v^{\mathrm{ref}}$ (left) and source term $f$ (right) in \ref{['ex:ex1']}. Red circle indicates the interface.

Theorems & Definitions (17)

• Theorem 3.1
• Example 5.1
• Example 5.2
• Example 5.3
• Lemma A.1
• proof
• Lemma A.2
• proof
• Proposition A.3
• proof