Numerical issues related to the solution of the Saint Venant equations of one-dimensional open -channel flow.

摘要

A von Neumann, or Fourier, stability analysis was applied to the numerical scheme currently used in a hydraulic model in order to determine whether the scheme is stable; that is, whether the difference between the exact solution and the numerical approximation of the governing Saint Venant equations grows or decays in a time-dependent process. The four-point implicit finite difference scheme, sometimes known as the Preissmann scheme, was also studied. The stability analysis was performed using a linearized form of the Saint Venant equations because rigorous procedures are not currently available for nonlinear equations. The resulting equations were solved for constant values of flow rate, cross-sectional area, and flow depth, and variable values of spatial flow increments, temporal increments, wave amplitude, spatial weighting coefficients, and temporal weighting coefficients. For different cases, the results confirmed that the four-point implicit scheme is unconditionally stable for different values of Δt/Δx, while maintaining the value of Φ (spatial weighting coefficient) equal to 0.5 (Preissmann scheme) for values of &thetas; (temporal weighting coefficient) near 0.60.;Pivoting was incorporated in the process of solving the linear system of equations that results after discretizing the Saint Venant equations using the four-point implicit scheme and applying the Newton-Raphson algorithm to the resulting set of nonlinear equations. Both exchange of rows only (partial pivoting) and exchange of rows and columns (full pivoting) were investigated. Partial pivoting was used with the LU (lower and upper) decomposition linear equation solver, while full pivoting was used with the Gauss-Jordan elimination algorithm. It was demonstrated that pivoting can, in some cases, prevent numerical divergence of the solution when simple Gaussian elimination would not.;A transformation of the Saint Venant equations was achieved with the intent of separating (decoupling) the dependent variables, such that each equation had only one dependent variable. This was done in preparation for the application of a TVD method as a possible solution to numerical instability problems that are sometimes encountered in models based on the original Saint Venant equations. Four different categories of numerical approaches to solving the transformed equations were developed and tested; all except one were unstable, even starting from a steady-state, uniform flow condition. The only stable algorithm involved a first-order “upwinding” and “downwinding” differencing of the transformed equations.;It was concluded that the four-point implicit method for the solution of the Saint Venant equations is most stable for high values of the Δt/Δx ratio, and weighting coefficients in the range of 0.5 to 0.6. The application of partial and full pivoting to the solution of the linear set of equations during Newton-Raphson iterations can make the difference between convergence and divergence of the solution, and should be applied as needed. However, full pivoting should be used only when needed because it slows the simulation down considerably. Finally, the transformed Saint Venant equations are highly unstable for most solution schemes, except for one of the explicit approaches that was attempted in this research.