Stochastic Modelling of 1-D Shallow Water Flows over Uncertain Topography

A second-order perturbation approach is used to investigate the effects of topographic uncertainty on a numerical model of shallow water flow. The governing equation is discretised using finite differences, the resulting nonlinear system expanded as a Taylor series about the unperturbed water depth to first and second-order, and the resulting matrix equation solved to derive second-order moments for the model solution. A Fourier technique is used to estimate the accuracy of the first- and second-order approximations and indicates that for even small perturbations, second-order terms are significant. Results are compared to those from Monte Carlo simulations, showing that significant nonlinear effects are well represented by the second-order stochastic model, predicting correctly the shift in the mean depth and an increase in the depth variance. The statistics of the solution are however still well represented by a Gaussian distribution, and therefore moments greater than order 2 need not be calculated.