Higher order accurate simulation of compressible multi-phase flows by means of a Discontinuous Galerkin method with non-smooth basis functions

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The numerical simulation of compressible multi-phase flows is extremely challenging for many numerical methods. Among other reasons, this is due to the inherent multi-scale character of the occurring solutions, the rapid movement of the sharp interface, the large jump in fluid properties and the presence of interfacial forces such as the surface tension. Recently, the Discontinuous Galerkin method has gained much attention in the context of various types of single-phase flows, especially because of the remarkably high convergence rates that can be achieved under very general conditions. However, existing extensions to multi-phase flows typically fall back to low convergence orders in the vicinity of the phase interface in order to improve the stability of the method and to avoid non-physical oscillations that inevitably occur if a discontinuous function is approximated by higher order polynomials. As a result, this project is targeted at overcoming these problems by introducing a cell-local, non-smooth enrichment into the polynomial approximation space. Since the location of the discontinuity is inferred from the zero iso-contour of a level set function, the construction of the enrichment is very simple and efficient. By virtue of a novel quadrature technique that avoids the necessity to reconstruct the interface explicitly, integrals over the induced sub-domains can be computed efficiently with hp-accuracy. At the same time, the introduction of the enrichment implies principal challenges, most notably in terms of stability and time-stepping schemes, which will be considered as key issues to be solved in the present project. First results from a related project where the aforementioned technique has been used in the context of incompressible multi-phase flows indicate that it is very well suited for overcoming the above-mentioned limitations. The mentioned project is based on the BoSSS framework which will also serve as a basis for the present project, thus allowing for a close cooperation. Within this project, we will refine the new methodology and apply it to flows comprising at least one compressible species. In particular, we are interested in the simulation of the collapse of isolated cavitation bubbles under the influence of surface tension. Experiments on a corresponding set-up will be performed by our cooperation partners and the results will serve as a basis for the verification of our results. Furthermore, our mid-term goal is the realization of a robust and extensible solver that can be used in follow-up projects.