Adaptive Q‐tree Godunov‐type scheme for shallow water equations

Abstract

This paper presents details of a second-order accurate, Godunov-type numerical model of the two-dimensional shallow water equations (SWEs) written in matrix form and discretized using finite volumes. Roe's flux function is used for the convection terms and a non-linear limiter is applied to prevent unwanted spurious oscillations. A new mathematical formulation is presented, which inherently balances flux gradient and source terms. It is, therefore, suitable for cases where the bathymetry is non-uniform, unlike other formulations given in the literature based on Roe's approximate Riemann solver. The model is based on hierarchical quadtree (Q-tree) grids, which adapt to inherent flow parameters, such as magnitude of the free surface gradient and depth-averaged vorticity. Validation tests include wind-induced circulation in a dish-shaped basin, two-dimensional frictionless rectangular and circular dam-breaks, an oblique hydraulic jump, and jet-forced flow in a circular reservoir. Copyright © 2001 John Wiley & Sons, Ltd.