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978-3-8439-0945-7, Reihe Mathematik
Numerical Methods for Sharp-Interface Dynamics of Phase Boundaries
98 Seiten, Dissertation Universität Stuttgart (2012), Softcover, A5
In this work we develop and study numerical methods for phase transition models. Phase transition models, as introduced here, can be used to describe phenomena such as liquid-vapour flows, the behaviour of metal alloys, and traffic flow. By phase transition model we mean a system of conservation laws of mixed hyperbolic-elliptic type. In this setting nonclassical waves are interpreted as phase boundaries. The behaviour of the phase boundary is described by a constitutive law - the so-called kinetic relation. Concerning the numerical treatment, classical methods like finite volume and discontinuous Galerkin are not directly applyable.
In the work at hand we propose two new methods for the above mentioned model based on Riemann solvers. The new methods can be introduced for systems of conservation laws in arbitrary space dimensions.
We prove convergence in a simplified setting and perform numerical experiments to validate the methods. As a further improvement, we replace the exact Riemann solver by a newly developed approximative Riemann solver. We detail the construction of the approximative Riemann solver for two specific cases. However, it can be also applied to other situations. This new approach reduces the computational cost significantly, which will be shown by numerical experiments. In our numerical experiments we show further simulations of applications, e.g., oscillating bubbles.