Mathematical and numerical study of a model for navigation in stratified waters

TL;DR

This paper develops and analyzes a linear spectral model for navigation in stratified waters, including stability, a critical speed regime, and a numerical scheme with proven error bounds, supported by numerical experiments.

Contribution

It introduces a spectral linear model with damping for stratified water navigation and provides theoretical analysis and numerical validation.

Findings

Existence and uniqueness of solutions under regularity conditions

Identification of a critical speed separating different regimes

Numerical experiments confirming theoretical results

Abstract

We derive a linear model of navigation in a two-layer fluid with a variable velocity of the ship. A spectral version of the model including a Rayleigh damping term is analyzed. We prove that the Cauchy problem has a unique solution if the velocity and if the initial data are sufficiently regular. The case of a constant speed is thoroughly investigated and the importance of a critical speed which separates two types of regimes is pointed out. We propose a numerical scheme based on the discrete Fourier transform for the space discretization and on an exponential integrator for the time discretization. We prove an error estimate for the exponential integrator. Numerical experiments in one and two space dimensions complete the theoretical results.