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

可压缩多相流的数值模拟对于许多数值方法来说都是极具挑战性的。在其他原因中，这是由于发生的溶液固有的多尺度特征、尖锐界面的快速移动、流体性质的大跳跃以及界面力(如表面张力)的存在。最近，间断Galerkin方法在各种类型的单相流中得到了广泛的关注，特别是因为在非常一般的条件下可以达到非常高的收敛速度。然而，现有的多相流扩展通常在相界面附近退回到低收敛阶，以提高方法的稳定性，并避免在用高阶多项式逼近不连续函数时不可避免地发生非物理振荡。因此，这个项目的目标是通过在多项式近似空间中引入单元局部的非光滑浓缩来克服这些问题。由于不连续的位置是从水平集函数的零等值线推断出来的，所以丰富的构造非常简单和有效。借助于一种新的求积技术，避免了显式重建界面的需要，可以高效地以hp精度计算诱导子域上的积分。同时，采用浓缩措施意味着面临主要挑战，尤其是在稳定性和时间步进计划方面，这些问题将被视为本项目要解决的关键问题。首先，将上述技术用于不可压缩多相流的一个相关项目的结果表明，该技术非常适合克服上述限制。上述项目是以BOSS框架为基础的，该框架也将作为本项目的基础，因此可以进行密切合作。在这个项目中，我们将改进新的方法，并将其应用于至少包含一个可压缩物种的流动。特别是，我们感兴趣的是孤立空化气泡在表面张力影响下坍塌的模拟。我们的合作伙伴将进行相应设置的实验，实验结果将作为验证我们结果的基础。此外，我们的中期目标是实现一个健壮且可扩展的求解器，可用于后续项目。

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.