Multiscale Problems in Three Applications - Abstract
Rohde, Christian
The dynamics of a compressible fluid which occurs in a liquid and a vapour phase can be described on the continuum mechanical level as a free boundary problem where the bulk phases are governed by the Euler equations (or the Navier-Stokes equations in viscous media). As conditions at the free boundary mass conservation, the dynamic Young-Laplace law and an additional relation, sometimes called kinetic relation, are enforced. The kinetic relation can be seen as the up-scaled information from the microscopic level. From the numerical point of view the main difficulty is now the correct consideration of the pointwise conditions at the boundary. We present a ghostfluid algorithm combined with an almost classical Riemann-solver based finite volume technique that has the potential to handle such situations. In the talk we will first discuss the analytical solution of the 1D-Riemann problem for the two-phase Euler equation which takes into account special kinetic relations at the phase boundary. Contrary to the classical gas dynamics for one-phase media the Riemann solution contains up to five single waves. The results rely on the work of Abeyaratne & Knowles and LeFloch et al. in the framework of elastodynamics. The Riemann solution is then a key stone to establish the ghostfluid/finite volume technique in the second part of the talk. We will present one- and twodimensional numerical simulations and a convergence proof for a simple but basic initial value problem. The third part of the talk will be devoted to a re-interpretation of the whole approach as a heterogeneous multi-scale method. In fact this different view allows to integrate the micro-scale information at the phase boundary in the numerical method not only in form of kinetic relations. It is quite naturally possible to integrate e.g. local computations with phase field models like the Navier-Stokes-Korteweg equations.