A Novel Multislope MUSCL Scheme for Solving 2D Shallow Water Equations on Unstructured Grids

Haiyong Xu,Xingnian Liu,Chao Liu,Sheng Huang,Fujian Li

doi:10.3390/w10040524

Abstract

Within the framework of the two-dimensional cell-centered Godunov-type finite volume (CCFV) method, this paper presents a novel multislope scheme on the basis of the monotone upstream scheme for conservation law (MUSCL) for numerically solving nonlinear shallow water equations on two-dimensional triangular grids. The Riemann states of the considered edge are calculated by an edge-based reconstructing procedure, where a limited scalar slope is employed to prevent potential numerical oscillations. The novel aspect of the new scheme is that it takes advantage of the geometrical characteristics of triangular grids in the reconstructing and limiting procedures, which effectively reduces the cost of computation and provides higher resolution and accuracy compared with classical MUSCL schemes. Seven tests are adopted to verify the scheme, and the results indicate that this scheme is efficient, accurate, robust, and high-resolution, and can be an ideal alternative for solving shallow water problems over uneven and frictional topography.

Highlights

Two-dimensional shallow water equations (SWEs) have been increasingly employed to simulate predominantly horizontal, free surface flows over natural topography, for instance, dam-break flow, urban flooding, and tsunami inundation
A new multislope monotone upstream scheme for conservation law (MUSCL) scheme on triangular grids is proposed in this paper within the framework of the Godunov-type cell-centered finite volume (CCFV) method for simulating 2D shallow water flows over uneven topography
The values of flow variables on both sides of the considered edge are calculated by an edge-based reconstructing and limiting procedure where the upwind slope is obtained, since the upwind point is fixed at the node corresponding to the considered edge

Summary

Introduction

Two-dimensional shallow water equations (SWEs) have been increasingly employed to simulate predominantly horizontal, free surface flows over natural topography, for instance, dam-break flow, urban flooding, and tsunami inundation. SWEs are a set of nonlinear hyperbolic-type partial differential equations, and several kinds of methods have been developed to solve them, such as the finite difference method (FDM) [3], the finite element method (FEM) [4], the finite volume method (FVM) [5], and the discontinuous Galerkin (DG). The Godunov-type scheme [8] within the framework of FVM, taking into account the propagation of flow information in resolving a local Riemann problem [9], is probably the most applied method of simulating free surface flows, for its inherent conservation property and stability in mixed flow regimes [10]. There are basically two approaches to derive FVM schemes, the cell-centered finite volume (CCFV) approach and the node-centered finite volume (NCFV) approach. The mixed CCFV and NCFV discretization method, which defines different flow variables separately at centroids with cells as control volumes, or at nodes with the

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Conclusion