A path conservative finite volume method for a shear shallow water model

TL;DR

This paper introduces a novel path conservative finite volume method for the shear shallow water model, incorporating vertical shear effects, and develops specialized solvers and schemes validated through test cases.

Contribution

It presents a new numerical scheme for the shear shallow water model with non-conservative structure, including HLL, HLLC-type solvers, and a second order MUSCL-Hancock scheme.

Findings

The method accurately captures shear effects in shallow water flows.

The proposed scheme performs well on test cases including roll waves.

It demonstrates improved stability and accuracy over classical approaches.

Abstract

The shear shallow water model provides an approximation for shallow water flows by including the effect of vertical shear in the model. This model can be derived from the depth averaging process by including the second order velocity fluctuations which are neglected in the classical shallow water approximation. The resulting model has a non-conservative structure which resembles the 10-moment equations from gas dynamics. This structure facilitates the development of path conservative schemes and we construct HLL, 3-wave and 5-wave HLLC-type solvers. An explicit and semi-implicit MUSCL-Hancock type second order scheme is proposed for the time integration. Several test cases including roll waves show the performance of the proposed modeling and numerical strategy.