A Stability notion for the viscous Shallow Water Discrete-Velocity Boltzmann Equations

TL;DR

This paper investigates the stability of viscous shallow water lattice Boltzmann equations with reduced gravity, highlighting the importance of parameter choice for maintaining stability in numerical simulations.

Contribution

It introduces a stability notion for viscous shallow water lattice Boltzmann equations and demonstrates the critical impact of reduced gravity on stability.

Findings

Reduced gravity significantly affects stability.

Careful parameter selection is essential for stable simulations.

Stability is validated through numerical tests.

Abstract

The stability of Lattice Boltzmann Equations modelling Shallow Water Equations in the special case of reduced gravity is investigated theoretically. A stability notion is defined as applied in incompressible Navier-Stokes equations in Banda, M. K., Yong, W.- A. and Klar, A: A stability notion for lattice Boltzmann equations. SIAM J. Sci. Comput. {\bf 27(6)}, 2098-2111 (2006). It is found that to maintain stability a careful choice of the value of the reduced gravity must be made. The stability notion is employed to investigate different shallow water lattice Boltzmann Equations. Results are tested using the Lattice Boltzmann Method for various values of the governing parameters of the flow. It is observed that even for the discrete model the reduced gravity has a significant effect on the stability.