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

2025/10/12 by Grajales, Juan Carlos Muñoz, Pizo, Deissy Marcela
#Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2510.10829

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 \citePizoMunoz 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.

Citations

• An inverse problem for linear system of dispersive equations
• A framework for automated PDE-constrained optimisation
• A Limited Memory Algorithm for Bound Constrained Optimization

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