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978-3-8439-1916-6, Reihe Mathematik

Michael Dudzinski
Well-Balanced Bicharacteristic-Based Finite Volume Schemes for Multilayer Shallow Water Systems

183 Seiten, Dissertation Technische Universität Hamburg-Harburg (2014), Softcover, A5

Zusammenfassung / Abstract

In this thesis Finite Volume Evolution Galerkin (FVEG) schemes for solving the classical shallow water as well as the two-layer shallow water equations both extended to treat the presence of a bottom topography as well as the influence of the Coriolis force are considered. Since these equations include nonconservative products the classical definition of weak solutions has to be extended for nonconservative systems. This extension is performed in the sense of the theory of paths.

In the second chapter a quick glance at Finite Volume schemes and reconstruction techniques is given and, in particular, the framework of the genuinely multidimensional FVEG schemes is explained. The FVEG framework relies on theory of bicharacteristics that is recalled quickly. Afterwards the FVEG framework is considered in the frame of path-conservative schemes.

Having set the groundwork, an earlier presented scheme for the shallow water equations is modified leading to an exactly well-balanced scheme for the jet in the rotational frame. This part requires an understanding of the steady states and the development of a new evolution operator. The scheme proposed is further extended to fit into path-conservative frame. Numerical results confirm the proven statements of exact well-balance.

The main part is presented in the last chapter and refers to the two-layer shallow water systems. These are deeply analyzed in terms of the eigenstructure. Having gained some insight into the geometry of the two-layer shallow water equations two schemes are developed. The first one, FVSEG, relies on a splitting technique. Here the layers are treated separately and thus the developed operator for the shallow water system is employed. Although this approach yields a feasible scheme it suffers from some limits that are demonstrated and, indeed, apply to any splitting approach. The second scheme, FVNEG, treats to whole system. Since the eigenstructure of the two-layer shallow water system is not readily available the evolution operator must be treated numerically at additional levels. The presented numerical treatment of the evolution operator is applicable to any hyperbolic system. The FVNEG scheme is then extended in a general manner to be able to deal with wet/dry fronts and the well-balanced character is shown even in the presence of dry states. In the section presetting the numerical results both schemes are compared and good reliability is presented.