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On the limiting amplitude principle for the wave equation with variable coefficients

Anton Arnold, Sjoerd Geevers, Ilaria Perugia, Dmitry Ponomarev

TL;DR

The paper analyzes the limiting amplitude principle (LAP) for the wave equation with spatially varying coefficients, not necessarily in divergence form, and establishes LAP in dimensions $d=2,3$ with a 1D extension. The authors develop a two-step approach: first transform the time-domain problem into a homogeneous auxiliary problem, then decompose the transformed problem into several subproblems whose time-decay is quantified using weighted-energy and propagator estimates. They obtain explicit convergence rates to the Helmholtz solution, showing algebraic decay in 2D/3D and exponential decay in 1D, and provide a detailed treatment of both localised initial data and locally supported sources. The results have practical implications for numerical methods that reformulate Helmholtz problems in the time domain and quantify modeling errors, with numerical illustrations supporting the theoretical decay rates.

Abstract

In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is extended to one space dimension with an appropriate modification. We also quantify the LAP and thus provide estimates for the convergence of the time-domain solution to the frequency-domain solution. Our proofs are based on time-decay results of solutions of some auxiliary problems. The obtained results are illustrated numerically on radially symmetric problems in dimensions 1, 2 and 3.

On the limiting amplitude principle for the wave equation with variable coefficients

TL;DR

The paper analyzes the limiting amplitude principle (LAP) for the wave equation with spatially varying coefficients, not necessarily in divergence form, and establishes LAP in dimensions with a 1D extension. The authors develop a two-step approach: first transform the time-domain problem into a homogeneous auxiliary problem, then decompose the transformed problem into several subproblems whose time-decay is quantified using weighted-energy and propagator estimates. They obtain explicit convergence rates to the Helmholtz solution, showing algebraic decay in 2D/3D and exponential decay in 1D, and provide a detailed treatment of both localised initial data and locally supported sources. The results have practical implications for numerical methods that reformulate Helmholtz problems in the time domain and quantify modeling errors, with numerical illustrations supporting the theoretical decay rates.

Abstract

In this paper, we prove new results on the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients, not necessarily in divergence form. Under suitable assumptions on the coefficients and on the source term, we establish the LAP for space dimensions 2 and 3. This result is extended to one space dimension with an appropriate modification. We also quantify the LAP and thus provide estimates for the convergence of the time-domain solution to the frequency-domain solution. Our proofs are based on time-decay results of solutions of some auxiliary problems. The obtained results are illustrated numerically on radially symmetric problems in dimensions 1, 2 and 3.
Paper Structure (14 sections, 12 theorems, 284 equations, 1 figure, 1 table)

This paper contains 14 sections, 12 theorems, 284 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Time-decay results
  3. Proof of the LAP (Theorems \ref{['thm:LAP']} and \ref{['thm:LAP_1D']})
  4. Step 1: Transformation into an auxiliary homogeneous problem.
  5. Slow decay of the initial data of problem \ref{['eq:wave_W']}
  6. Step 2: Time-decay by decomposition into subproblems.
  7. Proofs of the auxiliary time decay results
  8. Proof of Proposition \ref{['prop:dec_free']}
  9. Proof of Proposition \ref{['prop:dec_source']}
  10. Proof of Lemma \ref{['lem:gen_dec']}
  11. A variant of Lemma \ref{['lem:gen_dec']}
  12. Proof of Lemma \ref{['lem:spec_dec']}
  13. Conclusions and outlook
  14. Acknowledgements

Key Result

Theorem 1.4

Let $d=2, 3$. Suppose that Assumptions assm:alph_bet_base--assm:F_base are satisfied. Let $U\left(\mathbf{x}\right)$ and $u\left(\mathbf{x},t\right)$ be solutions to eq:U_Helm and eq:wave_LAP, respectively. Then, there exists a constant $C>0$ depending on $F$, $\alpha$, $\beta$, $\omega$, and $\Omeg for $d=3$: where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded domain.

Figures (1)

  • Figure 1: Large-time convergence for the radially symmetric example with the data as in \ref{['eq:alph_num']}--\ref{['eq:F_num']}.

Theorems & Definitions (18)

  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5: paper-decay1D, Prop. 1.1, Thm. 1.4
  • Lemma 4.1
  • proof
  • ...and 8 more