Numerical approximation of the 3d hydrostatic Navier-Stokes system with free surface

TL;DR

This paper introduces a stable numerical method for approximating the 3D hydrostatic Navier-Stokes system with free surface, enabling accurate simulation of complex fluid flows like tsunamis without moving meshes.

Contribution

It presents a novel Galerkin-based approximation and finite volume scheme for 3D hydrostatic Navier-Stokes equations with free surface, including a kinetic interpretation and stability analysis.

Findings

Accurate reproduction of tsunami wave propagation.

Stable and positivity-preserving numerical scheme.

Validation with analytical and realistic tsunami simulations.

Abstract

In this paper we propose a stable and robust strategy to approximate the 3d incompressible hydrostatic Euler and Navier-Stokes systems with free surface. Compared to shallow water approximation of the Navier-Stokes system, the idea is to use a Galerkin type approximation of the velocity field with piecewise constant basis functions in order to obtain an accurate description of the vertical profile of the horizontal velocity. Such a strategy has several advantages. It allows - to rewrite the Navier-Stokes equations under the form of a system of conservation lawswith source terms, - the easy handling of the free surface, which does not require moving meshes, - the possibility to take advantage of robust and accurate numerical techniques developed in extensive amount for Shallow Water type systems. Compared to previous works of some of the authors, the three dimensional case is studied in this paper. We show that the model admits a kinetic interpretation including the vertical exchanges terms, and we use this result to formulate a robust finite volume scheme for its numerical approximation. All the aspects of the discrete scheme (fluxes, boundary conditions,. . . ) are completely described and the stability properties of the proposed numerical scheme (well-balancing, positivity of the water depth,. . . ) are discussed. We validate the model and the discrete scheme with some numerical academic examples (3d non stationary analytical solutions) and illustratethe capability of the discrete model to reproduce realistic tsunami waves.