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Reflectionless discrete perfectly matched layers for higher-order finite difference schemes

Vicente A. Hojas, Carlos Pérez-Arancibia, Manuel A. Sánchez

TL;DR

The paper develops Reflectionless discrete PMLs (RDPMLs) for arbitrary high-order finite-difference discretizations of the scalar wave equation by embedding discrete complex stretching within a discrete holomorphic framework. It proves exponential decay of the holomorphic extension in the PML and demonstrates true, interface-level reflectionlessness across 1D and 2D test problems, including waveguides, with near-machine-precision absorption. The authors derive both time-domain full systems with auxiliary variables and frequency-domain reduced Helmholtz formulations, and analyze sparsity and solver implications. Numerical results show 2p-order convergence in 1D, strong dispersion control in 2D, and effective evanescent-wave handling via a two-stage stretching approach, highlighting practical benefits for accurate truncation of unbounded domains in high-order FD simulations.

Abstract

This paper introduces discrete-holomorphic Perfectly Matched Layers (PMLs) specifically designed for high-order finite difference (FD) discretizations of the scalar wave equation. In contrast to standard PDE-based PMLs, the proposed method achieves the remarkable outcome of completely eliminating numerical reflections at the PML interface, in practice achieving errors at the level of machine precision. Our approach builds upon the ideas put forth in a recent publication [Journal of Computational Physics 381 (2019): 91-109] expanding the scope from the standard second-order FD method to arbitrary high-order schemes. This generalization uses additional localized PML variables to accommodate the larger stencils employed. We establish that the numerical solutions generated by our proposed schemes exhibit an exponential decay rate as they propagate within the PML domain. To showcase the effectiveness of our method, we present a variety of numerical examples, including waveguide problems. These examples highlight the importance of employing high-order schemes to effectively address and minimize undesired numerical dispersion errors, emphasizing the practical advantages and applicability of our approach.

Reflectionless discrete perfectly matched layers for higher-order finite difference schemes

TL;DR

The paper develops Reflectionless discrete PMLs (RDPMLs) for arbitrary high-order finite-difference discretizations of the scalar wave equation by embedding discrete complex stretching within a discrete holomorphic framework. It proves exponential decay of the holomorphic extension in the PML and demonstrates true, interface-level reflectionlessness across 1D and 2D test problems, including waveguides, with near-machine-precision absorption. The authors derive both time-domain full systems with auxiliary variables and frequency-domain reduced Helmholtz formulations, and analyze sparsity and solver implications. Numerical results show 2p-order convergence in 1D, strong dispersion control in 2D, and effective evanescent-wave handling via a two-stage stretching approach, highlighting practical benefits for accurate truncation of unbounded domains in high-order FD simulations.

Abstract

This paper introduces discrete-holomorphic Perfectly Matched Layers (PMLs) specifically designed for high-order finite difference (FD) discretizations of the scalar wave equation. In contrast to standard PDE-based PMLs, the proposed method achieves the remarkable outcome of completely eliminating numerical reflections at the PML interface, in practice achieving errors at the level of machine precision. Our approach builds upon the ideas put forth in a recent publication [Journal of Computational Physics 381 (2019): 91-109] expanding the scope from the standard second-order FD method to arbitrary high-order schemes. This generalization uses additional localized PML variables to accommodate the larger stencils employed. We establish that the numerical solutions generated by our proposed schemes exhibit an exponential decay rate as they propagate within the PML domain. To showcase the effectiveness of our method, we present a variety of numerical examples, including waveguide problems. These examples highlight the importance of employing high-order schemes to effectively address and minimize undesired numerical dispersion errors, emphasizing the practical advantages and applicability of our approach.
Paper Structure (15 sections, 5 theorems, 111 equations, 13 figures, 2 tables)

This paper contains 15 sections, 5 theorems, 111 equations, 13 figures, 2 tables.

Table of Contents

  1. Introduction
  2. High-order finite-difference schemes in one dimension
  3. Discrete complex stretching
  4. RDPML equations in one dimension
  5. RDPML equations in two dimensions
  6. Frequency-domain reduced system
  7. Numerical results
  8. One-dimensional wave equation
  9. Two-dimensional wave equation
  10. Waveguide
  11. Evanescent waves
  12. Conclusions and ongoing work
  13. Continuous limit of the RDMPL equations
  14. Optimizing the damping factor $\sigma$
  15. Boundary conditions at RDPML boundaries

Key Result

Lemma 2.1

Let $\lambda\in\mathbb C$ and $\{z_r(\lambda)\}_{r=1}^p\subset\mathbb C$ be the roots of the $p$th-degree polynomial $P_p(\cdot;\lambda)\in \mathcal{P}^{p}(\mathbb C)$ defined as which are assumed to be distinct. Then, the general solution $0\neq \mathsf v\in \mathbb C^{\mathbb Z}$ of satisfies $\mathsf v\in E_\lambda$, where $E_\lambda\subset \mathbb C^\mathbb Z$ is defined as in terms of

Figures (13)

  • Figure 1: Discrete mode locations.
  • Figure 2: Depiction of the discrete complex domain $(\Lambda,\mathsf Z)$ defined by \ref{['eq:complex_dom']} and stretching path \ref{['eq:complex_path']}, which is marked by the red dots.
  • Figure 3: Comparison of the sparsity patterns of the full (a) and reduced (b) linear systems matrices resulting from the fourth-order RDPML scheme for the 1D time-harmonic wave (Helmholtz) equation. In (b) the $A_{11}$ matrix block is associated with the nodes in the physical domain, while the dense $A_{22}$ block corresponds to the nodes in the PML domain.
  • Figure 4: History of convergence of the approximate solutions using the RDPML schemes of order 2, 4, 6, and 8 for the one-dimensional wave equation. The errors are computed using the formula \ref{['eq:discrete_error']}, and the marks in the lines correspond to the values of $h$ used in the computations.
  • Figure 5: Time evolution of the numerical error $E_{\rm 1D}$\ref{['eq:discrete_error_2']} that measures the spurious reflections introduced by the proposed RDPML schemes, for FD spatial discretizations of order 2, 4, 6, and 8. (a) A PML domain of length 10 is used for which waves do return to the physical domain during the time interval $[0,10]$ consider in the plot. (b) A shorter PML domain of length 4 is used, for which waves return to the physical domain at $t=5$ due to the periodic boundary conditions \ref{['eq:per_bc_1d']} employed.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Definition 3.1: Discrete holomorphicity
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 6 more