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Perfectly matched layers in time domain. A simple two-dimensional error analysis

Kurt Bryan, Michael S. Vogelius

TL;DR

The paper addresses the challenge of quantifying time-domain accuracy of Perfectly Matched Layers (PML) for the 2D wave equation by deriving explicit time-domain error bounds for simulations on the unit disk. It develops a rigorous framework based on Fourier analysis and complex-coordinate stretching to construct an $R$-truncated PML, providing an error bound that decays like $e^{-2\alpha_0 R}$ and validating it with numerical experiments. A time-domain PML system with auxiliary variables is derived and discretized via a finite-element method, with detailed initial/boundary conditions, and numerical tests confirm the theoretical bounds and show exponential improvement with increased truncation radius, up to the limit set by discretization error. The results offer practical guidance for implementing time-domain PML in 2D simulations, ensuring reliable approximations to the full-space solution while operating on a finite computational domain.

Abstract

Perfectly Matched Layers (PML) has become a very common method for the numerical approximation of wave and wave-like equations on unbounded domains. This technique allows one to obtain accurate solutions while working on a finite computational domain, and the technique is relatively simple to implement. Results concerning the accuracy of the PML method have been obtained, but mostly with regard problems at a fixed frequency. In this paper we provide very explicit time-domain bounds on the accuracy of PML for the two-dimensional wave equation and illustrate our conclusions with some numerical examples.

Perfectly matched layers in time domain. A simple two-dimensional error analysis

TL;DR

The paper addresses the challenge of quantifying time-domain accuracy of Perfectly Matched Layers (PML) for the 2D wave equation by deriving explicit time-domain error bounds for simulations on the unit disk. It develops a rigorous framework based on Fourier analysis and complex-coordinate stretching to construct an -truncated PML, providing an error bound that decays like and validating it with numerical experiments. A time-domain PML system with auxiliary variables is derived and discretized via a finite-element method, with detailed initial/boundary conditions, and numerical tests confirm the theoretical bounds and show exponential improvement with increased truncation radius, up to the limit set by discretization error. The results offer practical guidance for implementing time-domain PML in 2D simulations, ensuring reliable approximations to the full-space solution while operating on a finite computational domain.

Abstract

Perfectly Matched Layers (PML) has become a very common method for the numerical approximation of wave and wave-like equations on unbounded domains. This technique allows one to obtain accurate solutions while working on a finite computational domain, and the technique is relatively simple to implement. Results concerning the accuracy of the PML method have been obtained, but mostly with regard problems at a fixed frequency. In this paper we provide very explicit time-domain bounds on the accuracy of PML for the two-dimensional wave equation and illustrate our conclusions with some numerical examples.

Paper Structure

This paper contains 14 sections, 6 theorems, 101 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Inhomogeneous wave equation on the plane
  3. A PML-modified solution
  4. An example
  5. Representation of the PML equations as a system in time-domain
  6. Initial and boundary conditions for the PML system (\ref{['vpmleq']})-(\ref{['upmleq']}) in time-domain
  7. Numerical Examples
  8. Analytical solutions to the wave equation on $\mathbb{R}^2$
  9. Experiment 1
  10. Experiment 2
  11. Experiment 3
  12. Appendix
  13. Some symmetries
  14. Details of the numerics

Key Result

Lemma 1

Let $u_{pml}(t,r,\theta)$ be defined as the inverse Fourier transform in time of the function Then for $r<R$ we conclude that $u_{pml}$ is real-valued. We will refer to this as the "$R$-truncated PML version" of the solution to the wave equation (wave-eq). In other words the $R$-truncated PML version of the solution to the wave equation (wave-eq) is given by $u_{pml}(t,r,\theta) = 2 \Re \lef for

Figures (5)

  • Figure 1: Left panel: graph of $I_1(R)$ for $0\leq R\leq 3$ with $L=3$. Right panel: graph of $I_2(R)$ for $0\leq R\leq 3$ with $L=3$.
  • Figure 2: Solution to PML system at time $t=5$ on region $0\leq r\leq 2$
  • Figure 3: Left panel: Solution to PML system at time $t=5$ on region $0\leq r\leq 1$. Right panel: Solution to inhomogeneous wave equation on $\mathbb{R}^2$ at time $t=5$ on region $0\leq r\leq 1$.
  • Figure 4: Left panel: supremum of error $|u(5,r,\theta)-u_{pml}(5,r,\theta)|$ for $0\leq r\leq 1$ as function of truncation radius $R$. Right panel: Logarithmic plot of this error.
  • Figure 5: PML error as a function of offset $c$.

Theorems & Definitions (9)

  • Remark 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Proposition 1
  • Remark 3
  • Lemma 3
  • Lemma 4
  • Lemma 5