Solution of the Unsteady One-Dimensional Equations of Non-Linear Shallow Water Theory by the Lax-Wendroff Method, with Applications to Hydraulics

摘要

An adaptation of the two-step Lax-Wendroff method is used for solving the unsteady one-dimensional equations of non-linear shallow water theory, including both frictional resistance and lateral inflow terms. This finite difference method is fast, accurate and simple to program and covers the formation and subsequent history of discontinuities in the solution, in the form of bores and hydraulic jumps, without any special procedures. The behaviour of the numerical solution behind these jumps is found in the examples to be sufficiently smooth without the addition of an artificial viscous force term. A variety of illustrative examples is given, including simple cases of flood waves in rivers, bores in channels resulting from rapid changes of upstream conditions, oscillatory waves on a super-critical stream and a simple hydrology example with a significant lateral inflow from rain. Several checks of the numerical method are included. The examples are confined to channels of uniform rectangular cross-section, but the method generalises in a straightforward way to real rivers and estuaries in which the cross-section is non-rectangular and varies along the length of the channel. (Author)