ISBN 9783868538212

978-3-86853-821-2, Reihe Mathematik

Andreas Bollermann
Numerical Methods for the Shallow Water Equations

154 Seiten, Dissertation Rheinisch-Westfälische Technische Hochschule Aachen (2011), Softcover, A5

Zusammenfassung / Abstract

Our primary objective has been to extend the range of possible applications for the FVEG method. We successfully developed the following enhancements: 1. A FVEG method on adaptive grids was derived that preserves conserva- tion and well-balancing. 2. We presented a new entropy fix relying only on the correct choice of the sonic cone. 3. We developed a technique to use the FVEG method in the presence of dry areas. This includes the correct treatment of vacuum as well as the proper discretisation of the source term.

The adaptive FVEG method from Chapter 4 was successfully applied to the SWE. Although we already saw a gain in computational efficiency, we merely consider this a proof of concept. The use of adaptive techniques is mandatory for 3D problems, but for our test cases, the computational times were typically small enough to allow for uniform computations as is shown in Chapter 7. Further, the solutions to many of our test cases are not very localised, see e.g. the perturbation over a smooth bed in Sections 4.3 and 7.5.2. This is often different for, say, the Euler equations, where many test cases are shock dominated. However, the steps leading to an adaptive scheme easily transfer to 3D or other models, so we consider our contribution as valuable.

The new entropy fix is surprisingly effective, and fits very nicely into the concept of the EG schemes. But it brought up another aspect which could turn out to be even more important for the further development of the FVEG method. Until now, the influence of the choice of the linearisation and the particular approximation of EG operators for non-linear systems has not been thoroughly analysed. We pointed out in Sections 8.1 and 8.2 that the simple transfer of the linear strategies to the non-linear case does not necessarily carry over the desired properties of the scheme. From the entropy fix it is evident that these choices can be crucial for an accurate approximation of certain numerical problems, so there are some challenging questions for the future.