A Novel PML-type Technique for Acoustic Scattering Problems based on A Real Coordinate Transformation

TL;DR

The paper introduces Real Compressed Layer (RCL), a PML-type domain truncation for the exterior Helmholtz problem based on a real coordinate transformation that compresses the unbounded exterior into a bounded layer. By extracting the oscillatory factor from the transformed solution, the method yields a PDE with real coefficients and enables direct recovery of the far-field when mapped back, avoiding artificial, complex-stretched media. The authors establish exponential convergence bounds and demonstrate the approach on circular and rectangular layers, showing robust, non-reflective behavior and spectral-like accuracy using Fourier and spectral-element discretizations. The framework extends to both circular and rectangular truncations, with promising avenues for 3D and time-domain extensions, and offers practical advantages in terms of real-valued computations and direct far-field reconstruction.

Abstract

It is known that any {\em real coordinate transformation} (RCT) to compress waves in an unbounded domain into a bounded domain results in infinite oscillations that cannot be resolved by any grid-based method. In this paper, we intend to show that it is viable if the outgoing waves are compressed along the radial direction and the resulting oscillatory pattern is extracted explicitly. We therefore construct a perfectly matched layer (PML)-type technique for domain reduction of wave scattering problems using RCT, termed as real compressed layer (RCL). Different from all existing approaches, the RCL technique has two features: (i) the RCL-equation only involves real-valued coefficients, which is more desirable for computation and analysis; and (ii) the layer is not ``artificial'' in the sense that the computed field in the layer can recover the outgoing wave of the original scattering problem in the unbounded domain. Here we demonstrate the essential idea and performance of the RCL for the two-dimensional Helmholtz problem with a bounded scatterer, but this technique can be extended to three dimensions in a similar setting.

Figures (7)

• Figure 2.1: Schematic illustration of the circular RCL and PML domains.
• Figure 2.2: Profiles of the real part of $U(\rho,\theta), u(r,\theta), v(r,\theta)$ with $\theta=0,\pi/4$.
• Figure 2.3: The numerical solution in 2D for $\theta = 0, k = 50$, under the real mapping and polynomial order is $N_1=N_2=150.$
• Figure 3.1: Schematic illustrations of the rectangular PML and RCL with four trapezoidal patches, and the compression transformation from radial direction..
• Figure 3.2: 2D plots of numerical and exact solutions with $k = 50, N = 4.$

Theorems & Definitions (7)

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