Hyperbolicity of a semi-Lagrangian formulation of the hydrostatic free-surface Euler system

TL;DR

This paper reformulates the hydrostatic Euler system using a semi-Lagrangian approach, analyzes its hyperbolic structure, and proposes a numerical discretization method to solve and study the system's stability and solutions.

Contribution

It introduces a semi-Lagrangian formulation of the hydrostatic Euler equations, analyzes the hyperbolic structure including spectrum and invariants, and proposes a multilayer discretization method.

Findings

Spectrum includes continuous part, affecting stability analysis.

Riemann invariants identified as conserved quantities.

Discretization eigenvalues analyzed for hyperbolicity.

Abstract

By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations describing free-surface sheared flows is rewritten as a system of quasilinear equations, where stability conditions can be determined by the analysis of its hyperbolic structure. The system one obtains can be written as a quasi linear system in time and horizontal variables and involves no more vertical derivatives. However, the coefficients in front of the horizontal derivatives include an integral operator acting on the new vertical variable. The spectrum of these operators is studied in detail, in particular it includes a continuous part. Riemann invariants are then determined as conserved quantities along the characteristic curves. Examples of solutions are provided, in particular stationary solutions and solutions blowing-up in finite time. Eventually, we propose an exact multilayer $\mathbb{P}_0$-discretization, which could be used to solve numerically this semi-Lagrangian system, and analyze the eigenvalues of the corresponding discretized operator to investigate the hyperbolic nature of the approximated system.