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Analysis of linear Boussinesq-type models coupled with static interfaces

José Galaz, Maria Kazolea, Antoine Rousseau

TL;DR

This work analyzes the coupling of linearized Boussinesq-type and Saint-Venant shallow-water equations under a static interface at $x=0$, introducing a one-way reference model to derive an analytical solution and to study the full hybrid system. It proves Hadamard well-posedness for the coupled problem, and defines a coupling error that corresponds to interface reflections, shown to be $O(\mu^2)$ for small dispersion parameter $\mu$ in a broad data class. The analysis derives exact transmission conditions, characterizes the frequency-dependent reflection via $r(s)$, and provides explicit expressions for the reflected perturbations in both BSV and SVB configurations, linking the coupling error to the observable spurious oscillations reported in literature. The framework uses harmonic representations and rigorous trace/approximation theory to decompose and quantify the error without requiring a fully 3D Euler reference, enabling straightforward computation of the coupling error in varied scenarios. These results offer a principled basis for robust heterogeneous domain coupling in shallow-water modeling and can guide the design of improved transmission conditions for BT–SV hybrids.

Abstract

We derive a new approach to analyze the coupling of linear Boussinesq and Saint-Venant shallow water wave equations in the case where the interface remains at a constant position in space. We propose a one-way coupling model as a reference, which allows us to obtain an analytical solution, prove the well-posedness of the original coupled model and compute what we call the coupling error-a quantity that depends solely on the choice of transmission conditions at the interface. We prove that this coupling error is asymptotically small for a certain class of data and discuss its role as a proxy for the full error with respect to the 3D water wave problem. Additionally, we highlight that this error can be easily computed in other scenarios. We show that the coupling error consists of reflected waves and argue that this explains some previously unexplained spurious oscillations reported in the literature. Finally, we prove the well-posedness of the half-line linear Boussinesq problem.

Analysis of linear Boussinesq-type models coupled with static interfaces

TL;DR

This work analyzes the coupling of linearized Boussinesq-type and Saint-Venant shallow-water equations under a static interface at , introducing a one-way reference model to derive an analytical solution and to study the full hybrid system. It proves Hadamard well-posedness for the coupled problem, and defines a coupling error that corresponds to interface reflections, shown to be for small dispersion parameter in a broad data class. The analysis derives exact transmission conditions, characterizes the frequency-dependent reflection via , and provides explicit expressions for the reflected perturbations in both BSV and SVB configurations, linking the coupling error to the observable spurious oscillations reported in literature. The framework uses harmonic representations and rigorous trace/approximation theory to decompose and quantify the error without requiring a fully 3D Euler reference, enabling straightforward computation of the coupling error in varied scenarios. These results offer a principled basis for robust heterogeneous domain coupling in shallow-water modeling and can guide the design of improved transmission conditions for BT–SV hybrids.

Abstract

We derive a new approach to analyze the coupling of linear Boussinesq and Saint-Venant shallow water wave equations in the case where the interface remains at a constant position in space. We propose a one-way coupling model as a reference, which allows us to obtain an analytical solution, prove the well-posedness of the original coupled model and compute what we call the coupling error-a quantity that depends solely on the choice of transmission conditions at the interface. We prove that this coupling error is asymptotically small for a certain class of data and discuss its role as a proxy for the full error with respect to the 3D water wave problem. Additionally, we highlight that this error can be easily computed in other scenarios. We show that the coupling error consists of reflected waves and argue that this explains some previously unexplained spurious oscillations reported in the literature. Finally, we prove the well-posedness of the half-line linear Boussinesq problem.

Paper Structure

This paper contains 21 sections, 24 theorems, 212 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Boussinesq equations
  4. On the real line
  5. On the half line
  6. Saint-Venant equations
  7. Useful analysis theorems
  8. Trace theorems
  9. Approximation theory
  10. Analysis of coupled models
  11. The reference solution
  12. Characterization of the coupling conditions
  13. Analytical solution
  14. Boussinesq to Saint-Venant case
  15. Saint-Venant to Boussinesq case
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let $r\geq 0$, if $\bm{W}_0 =(\eta_0, u_0)\in H^r(\Omega) \times H^{r+1}(\Omega)$ then the Cauchy problem eq:lgn-homogeneous-cauchy has a unique solution $\bm{W} =(\eta, u) \in L^\infty(0,\infty; H^r(\Omega)\times H^{r+1}(\Omega))$ and

Figures (2)

  • Figure 1: Sketch of variables and scales.
  • Figure 2: Sketch of the approximation used to handle the case of initial data with arbitrary support. The original initial condition $\bm{W}_0$ (top figure) is approximated as the superposition of two initial conditions $\bm{W}_0^L$, $\bm{W}_0^R$ that are supported on the left and on the right of the interface respectively.

Theorems & Definitions (45)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • ...and 35 more