Two‐Dimensional Surges and Shocks in Open Channels

Abstract

The finite element method based on the classical Galerkin formulation produces very poor results when applied to discontinuous channel flow, although the complex geometry of most practical problems makes the use of finite elements very desirable. A variance of the Galerkin scheme for conservation laws in two‐dimensional, nearly horizontal flow, which exhibits a remarkable shock‐capturing ability, is presented. The parasitic waves in the vicinity of the discontinuity commonly present in the Gelerkin solution are selectively dissipated, and, in fact, the sharpness of the front is improved by the addition of the dissipation mechanism. The method is based on discontinuous weighting functions which introduce upwind effects in the solution while maintaining central difference accuracy. No arbitrary parameters are needed because the dissipation level is selected analytically. No higher order derivatives appear in the governing equations which simplifies the construction and execution of the scheme. Results are presented for several flow situations that give rise to spontaneous formation of surge and shock waves in two space dimensions. The accuracy of computation is verified by comparison with analytical solutions and by continuous monitoring of the conservation properties of the model.