Existence and numerical approximation of a one-dimensional Boussinesq system with variable coefficients on a finite interval

TL;DR

The work analyzes a one-dimensional Boussinesq-type system with spatially varying depth on a finite interval, addressing well-posedness, energy conservation, and numerical approximation. It uses Green's-function-based integral reformulations and Banach fixed-point arguments to establish local existence and uniqueness, plus an energy conservation law $E(t)=E(0)$. The authors implement and test a finite-element solver on two depth-variation profiles and formulate an inverse problem to reconstruct initial conditions from final-time data using adjoint-based optimization with L-BFGS-B. This study advances both the analytical understanding and numerical treatment of dispersive waves in variable-depth channels and demonstrates a foundation for data-assimilation approaches in such systems.

Abstract

In this paper, we investigate the well-posedness of a nonlinear dispersive model with variable coefficients that describes the evolution of surface waves propagating through a one-dimensional shallow water channel of finite length with irregular bottom topography. To complement the theoretical analysis, we utilize the numerical solver developed by the authors in \cite{PizoMunoz} to approximate solutions of the model on a finite spatial interval, considering various parameter values and forms of the variable coefficients in the Boussinesq system under study. Additionally, we present preliminary numerical experiments addressing an inverse problem: the reconstruction of the initial wave elevation and fluid velocity from measurements taken at a final time. This is achieved by formulating an optimization problem in which the initial conditions are estimated as minimizers of a functional that quantifies the discrepancy between the observed final state and the numerical solution evolved from a trial initial state.

Figures (5)

• Figure 1: Coefficients $c(\xi)$ used in the simulations.
• Figure 2: Wave elevation $N(\xi,t)$ and fluid velocity $V(\xi,t)$ at $t = 8$, for the coefficient $c(\xi)$ shown in Figure \ref{['coefficients']}(a).
• Figure 3: Wave elevation $N(\xi,t)$ and fluid velocity $V(\xi,t)$ at $t = 8$ for the coefficient $c(\xi)$ shown in Figure \ref{['coefficients']}(b).
• Figure 4: Approximation of the initial wave elevation $N_0(\xi)$ after the first four iterations of the L-BFGS-B algorithm used to minimize the functional \ref{['Functional_assimilation']}.
• Figure 5: Approximation of the initial fluid velocity $V_0(\xi)$ after the first four iterations of the L-BFGS-B algorithm used to minimize the functional \ref{['Functional_assimilation']}.

Theorems & Definitions (4)

• Theorem 2.1
• Lemma 3.1
• Theorem 3.1
• Theorem 3.2