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Integral equations for flexural-gravity waves: analysis and numerical methods

Travis Askham, Jeremy G. Hoskins, Peter Nekrasov, Manas Rachh

TL;DR

This work addresses the scattering of flexural-gravity waves by a thin plate with spatially varying thickness over deep water, a model relevant to sea ice and ice shelves. The authors formulate a two-step integral-equation framework: (i) reduce the 3D boundary value problem to a surface integro-differential equation on z=0, and (ii) specialize to a compactly supported perturbation using a Green function for the constant-coefficient problem, yielding a Fredholm second-kind equation for a surface density $\sigma$. They establish mapping and existence/uniqueness results, show high-order convergence for a Nyström discretization with 6th-order Zeta-corrected quadrature, and accelerate the FFT-based matrix-vector products to solve large-scale problems with complex geometries. The method is demonstrated on diverse geometries (random thickness, rolls, ridges, spirals) and shows accurate, scalable performance, enabling detailed exploration of wave-ice interactions and potential extensions to finite depth and non-smooth coefficients.

Abstract

In this work, we develop a fast and accurate method for the scattering of flexural-gravity waves by a thin plate of varying thickness overlying a fluid of infinite depth. This problem commonly arises in the study of sea ice and ice shelves, which can have complicated heterogeneities that include ridges and rolls. With certain natural assumptions on the thickness, we present an integral equation formulation for solving this class of problems and analyze its mathematical properties. The integral equation is then discretized and solved using a high-order-accurate, FFT-accelerated algorithm. The speed, accuracy, and scalability of this approach are demonstrated through a variety of illustrative examples.

Integral equations for flexural-gravity waves: analysis and numerical methods

TL;DR

This work addresses the scattering of flexural-gravity waves by a thin plate with spatially varying thickness over deep water, a model relevant to sea ice and ice shelves. The authors formulate a two-step integral-equation framework: (i) reduce the 3D boundary value problem to a surface integro-differential equation on z=0, and (ii) specialize to a compactly supported perturbation using a Green function for the constant-coefficient problem, yielding a Fredholm second-kind equation for a surface density . They establish mapping and existence/uniqueness results, show high-order convergence for a Nyström discretization with 6th-order Zeta-corrected quadrature, and accelerate the FFT-based matrix-vector products to solve large-scale problems with complex geometries. The method is demonstrated on diverse geometries (random thickness, rolls, ridges, spirals) and shows accurate, scalable performance, enabling detailed exploration of wave-ice interactions and potential extensions to finite depth and non-smooth coefficients.

Abstract

In this work, we develop a fast and accurate method for the scattering of flexural-gravity waves by a thin plate of varying thickness overlying a fluid of infinite depth. This problem commonly arises in the study of sea ice and ice shelves, which can have complicated heterogeneities that include ridges and rolls. With certain natural assumptions on the thickness, we present an integral equation formulation for solving this class of problems and analyze its mathematical properties. The integral equation is then discretized and solved using a high-order-accurate, FFT-accelerated algorithm. The speed, accuracy, and scalability of this approach are demonstrated through a variety of illustrative examples.
Paper Structure (22 sections, 30 theorems, 121 equations, 8 figures, 1 table)

This paper contains 22 sections, 30 theorems, 121 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem
  3. Reduction to an integral equation
  4. The constant coefficient problem
  5. Variable coefficients
  6. Analytical properties of the integral equation
  7. Mapping properties of the integral equation
  8. Properties of the solutions of the integral equation
  9. Existence and uniqueness of solutions of the integral equations
  10. Numerical implementation and fast algorithms
  11. Discretization, quadrature rule, and fast solution
  12. Convergence of the numerical method
  13. Examples and applications
  14. Conclusion
  15. Code availability
  16. ...and 7 more sections

Key Result

Lemma 3.1

Let $\rho_1,\ldots,\rho_5$ denote the roots of the polynomial If $\Im({\beta_0})\ne 0$, then none of the $\rho_j$ are real. If ${\beta_0} \in \mathbb{R}$, then exactly one of the $\rho_j$ is a positive real number, which we label by $\rho_1$. Supposing that all of the roots are distinct, we define the coefficients $e_1,\ldots,e_5$ by Additionally, the coefficients $e_j$ and roots $\rho_j$ satisfy

Figures (8)

  • Figure 1: Illustration of the problem setup.
  • Figure 2: Convergence of discretized integral operators from equation \ref{['eq:LS']} computed using trapezoid rule with 6th-order Zeta corrections.
  • Figure 3: Convergence of the solution to the integral equation \ref{['eq:LS']} for a Gaussian thickness profile and incident plane wave. The consistency of the solution $\phi$ with the surface PDE \ref{['eq:phipde2']} is shown in red and blue, while the self-convergence of the density $\mu$ is shown in yellow and purple.
  • Figure 4: A point source inside of a random flexural medium. Top left: thickness profile generated by a sum of Gaussian distributions with random height values. The average thickness is 5 m. Top right: the absolute value of the density $\mu$. This density can be viewed as the strength of sources that must be added to the original point source to produce the total field. Bottom: the absolute value of the velocity potential for a point source radiating outwards from the center of the domain.
  • Figure 5: Top: surface roll pattern with a wavelength of 333 m. Arrows indicate direction of the incident plane wave. The blue dot indicates where the scattered field was measured as a proxy for the amplitude of the reflected wave, while the red dot marks the location where the total field was measured for the amplitude of the transmitted wave. Bottom: amplitudes of the reflected and transmitted fields for a wide range of wavenumbers. Dashed lines represent wavenumbers values that are plotted in \ref{['fig:rolls2']}.
  • ...and 3 more figures

Theorems & Definitions (51)

  • Lemma 3.1
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Corollary 4.1
  • Corollary 4.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3: Rellich
  • ...and 41 more