TVD schemes for open channel flow

Abstract

SUMMARY The Saint Venant equations for modelling flow in open channels are solved in this paper, using a variety of total variation diminishing (TVD) schemes. The performance of second- and third-order-accurate TVD schemes is investigated for the computation of free-surface flows, in predicting dam-breaks and extreme flow conditions created by the river bed topography. Convergence of the schemes is quantified by comparing error norms between subsequent iterations. Automatically calculated time steps and entropy corrections allow high CFL numbers and smooth transition between different conditions. In order to compare different approaches with TVD schemes, the most accurate of each type was chosen. All four schemes chosen proved acceptably accurate. However, there are important differences between the schemes in the occurrence of clipping, overshooting and oscillating behaviour and in the highest CFL numbers allowed by a scheme. These variations in behaviour stem from the different orders and inherent properties of the four schemes. © 1998 John Wiley & Sons, Ltd. In this paper, numerical schemes for solving the Saint Venant equations, which model bulk channel flow, are presented. This set of hyperbolic equations yields discontinuous solutions, which can be difficult to represent accurately without the use of a modern shock-capturing method. There are problems with the high level of truncation errors when using first-order upwind schemes and with the oscillatory behaviour of most higher-order schemes. However, coupling a total variation diminishing (TVD) interpolation with an appropriate Riemann solver, yields a high-order-accurate scheme which numerically captures discontinuities with sharp corners and avoids unrealistic oscillations. The myriad of TVD schemes may be categorised into algebraic and geometric approaches. Further subdivisions of the algebraic schemes are symmetric, upwind and predictor-corrector. One scheme has been chosen from each of these forms of TVD schemes. The schemes used were a second-order symmetric, an upwind scheme called the modified flux, the two step TVD‐McCormack and the third-order MUSCL representing the geometric approach. Comparison of numerical schemes has a long tradition; here we summarise our experience with these numerical schemes and suggestions for further work are provided.