Flood forecasting and delineation;​ the design of flood protection works;​ and the assessment of the impact of dam failure or a sudden ice jam release, are all examples where knowledge of the relationships between depth and discharge in an open channel are required. However, even for a one-dimensional problem, the governing non-linear system of partial differential equations (known as the St. Venant equations) are extremely difficult to solve. This study provides a comprehensive examination of the state of the art in finite element modeling of open channel flow. The objective was to determine if a numerical technique exists, or could be developed, which would provide accurate and stable solutions for a wide range of one-dimensional open channel flow problems.

The finite element method was used for this study because of its underlying consistency and generality which allows for quick comparisons between numerical schemes. In addition, its geometric flexibility leads to efficient definition of the irregular features common to natural channels and boundary conditions are easily implemented. The schemes investigated were the Taylor-Galerkin;​ the Dissipative-Galerkin (DG);​ and the Least-Squares methods. In addition, a reconsideration of the fundamental role of the characteristics in the determination of the upwind weighting, has lead to a new approach entitled the Characteristic-Dissipative-Galeridn (CDG) scheme. Results were compared with the Bubnov-Galerkin and a four-point implicit finite difference scheme. Initially, a linear stability analysis was used to examine the amplification and phase characteristics of these methods. They were then tested extensively for both steady and unsteady flow situations including kinematic, diffusive and shallow water gravity waves.

By employing upwinded elements both the DG and CDG schemes were able to achieve superior performance compared to other methods, propagating disturbances without significant damping of the peak and displaying good phase accuracy. Both methods proved stable near shocks, although the CDG did not require an increase in the upwinding parameter.

It was concluded that the success of the CDG scheme in modeling a wide variety of problems without significant parameter variation, was due to its consideration of the fundamental role of the characteristics in quantifying the upwind weighting. This method could be extended to other hyperbolic systems such as:​ two-dimensional flows;​ multi-layer fluids;​ or sediment transport problems.