Quantitative Analysis of the Limiting Amplitude Principle for the Variable-Coefficient Wave Equation

Core Concepts

The authors prove the validity of the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients in spatial dimensions 1, 2, and 3. They quantify the convergence rates of the time-domain solution to the frequency-domain solution.

Abstract

The paper studies the limiting amplitude principle (LAP) for the wave equation with nonconstant coefficients. The authors consider the following setup:

Frequency-domain problem:

The Helmholtz equation with variable coefficients and a compactly supported source term.

Time-domain problem:

The wave equation with variable coefficients and a time-harmonic source term.
The authors make the following key assumptions:

Smoothness, positivity, and compact support of the variable coefficients.
Non-trapping condition on the coefficients (for d ≥ 2).
Compact support of the source term.
The main results are:

For d = 2, 3, the authors prove the validity of the LAP and establish algebraic convergence rates of the time-domain solution to the frequency-domain solution.

For d = 1, the authors prove the validity of the LAP with an appropriate modification and establish exponential convergence.

The proofs rely on time-decay estimates for solutions of auxiliary problems, which are established using a decomposition approach. The authors avoid a direct study of the resolvent operator and instead leverage recent results on time-decay of solutions.

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Key Insights Distilled From

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

by Anton Arnold... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2202.10105.pdf

Deeper Inquiries

How can the results be extended to more general classes of variable coefficients, such as those that are not necessarily smooth or compactly supported

To extend the results to more general classes of variable coefficients, such as those that are not necessarily smooth or compactly supported, we can consider relaxing the regularity and localization assumptions on the coefficients and source terms. This extension would involve adapting the decay estimates and convergence analysis to accommodate the increased complexity and variability in the coefficients. By employing more general function spaces and possibly incorporating techniques from functional analysis, such as distribution theory, we can address a broader range of coefficient functions with varying regularity and support properties.

Can the convergence rates obtained in the 3D case be further improved, potentially matching the 2D case

While the convergence rates obtained in the 3D case are already quite strong, there is potential for further improvement to match the rates achieved in the 2D case. One approach to enhancing the convergence rates in the 3D case could involve refining the analysis of the decay estimates for the solutions of the auxiliary problems. By exploring more intricate decay properties and leveraging advanced mathematical tools, it may be possible to derive sharper estimates that lead to faster convergence rates in the 3D scenario. Additionally, optimizing the decomposition strategies and refining the quantification techniques could contribute to achieving convergence rates on par with the 2D case.

What are the implications of the LAP results for the numerical analysis of time-domain methods for solving Helmholtz problems at high frequencies

The LAP results have significant implications for the numerical analysis of time-domain methods used to solve Helmholtz problems at high frequencies. By establishing the validity of the limiting amplitude principle and providing estimates for the convergence of time-domain solutions to frequency-domain solutions, the LAP offers a theoretical foundation for understanding the behavior of wave equations with variable coefficients. In the context of numerical methods, the LAP can guide the development of efficient algorithms for solving Helmholtz problems, particularly at high frequencies where traditional approaches may face challenges. The quantification of large-time convergence provided by the LAP can inform the design and optimization of numerical schemes, potentially leading to improved accuracy and computational efficiency in solving wave equations with variable coefficients.