Numerical Techniques for the Shallow Water Equations

In this report we will discuss some numerical techniques for approximating the Shallow Water equations. In particular we will discuss finite difference schemes, adaptations of Roe’s approximate Riemann solver and the Q-Schemes of Bermudez & Vazquez with the objective of accurately approximating the solution of the Shallow Water equations. We consider four different test problems for the Shallow Water equations with each test problem making the source term more significant, i.e. the variation of the Riverbed becomes more pronounced, so that the different approaches discussed in this report can be rigorously tested. A comparison of the different approaches discussed in this report will also be made so that we may determine which approach produced the most accurate numerical results overall. The work contained in this report has been carried out as part of the Oxford / Reading Institute for Computational Fluid Dynamics and was funded by the Engineering and Physical Science Research Council and HR Wallingford under a CASE award.