Asymptotic Analysis of Shallow Water Moment Equations

TL;DR

This paper performs an asymptotic analysis of the Shallow Water Moment Equations to derive a reduced model that is computationally more efficient and more accurate than traditional models for free-surface flows.

Contribution

It introduces the Reduced Shallow Water Moment Equations (RSWME) derived via asymptotic analysis, simplifying the SWME while maintaining accuracy near equilibrium states.

Findings

RSWME reduces computational cost by up to 77%.

RSWME improves accuracy over SWE by up to 88%.

The hyperbolicity of RSWME is analyzed.

Abstract

The Shallow Water Moment Equations (SWME) are an extension of the Shallow Water Equations (SWE) for improved modelling of free-surface flows. In contrast to the SWE, the SWME incorporate vertical velocity profile information. The SWME framework approximates vertical velocity profiles using a polynomial expansion with Legendre polynomials and polynomial coefficients, also called moment variables. The SWME have an increased number of variables that must always be incorporated, even when the flow approaches a viscous slip equilibrium state that could be characterised by vanishing moment variables. To reduce the complexity of the SWME in cases proximate to this equilibrium, we conduct an asymptotic analysis of the SWME. This yields the closed form Reduced Shallow Water Moment Equations (RSWME) for deviations from the equilibrium. The RSWME have fewer variables, compared to the SWME. The hyperbolicity of the RSWME is analysed. Numerical tests include a wave with a sharp height gradient, a smoother height gradient and a square root velocity profile. The numerical tests demonstrate that the RSWME reduce computational cost up to 77% compared to the SWME and improves accuracy up to 88% over the SWE.