Discrete Boltzmann model of shallow water equations with polynomial equilibria

TL;DR

This paper develops a discrete Boltzmann model for shallow water flows using Hermite expansion, analyzing convergence, conservation, and numerical validation to improve simulation of supercritical flows.

Contribution

It introduces a novel discrete Boltzmann model based on polynomial equilibria for shallow water equations, with analysis of convergence and numerical validation.

Findings

Convergence behavior of the expansion is complex but conservation laws are preserved.

Higher order expansion and quadrature improve simulation of supercritical flows.

The model effectively balances source and flux terms for steady solutions.

Abstract

A type of discrete Boltzmann model for simulating shallow water flows is derived by using the Hermite expansion approach. Through analytical analysis, we study the impact of truncating distribution function and discretizing particle velocity space. It is found that the convergence behavior of expansion is nontrivial while the conservation laws are naturally satisfied. Moreover, the balance of source terms and flux terms for steady solutions is not sacrificed. Further numerical validations show that the capability of simulating supercritical flows is enhanced by employing higher order expansion and quadrature.