In this work, we present a novel hybrid Discontinuous Galerkin scheme with hp-adaptivity capabilities for the compressible Euler equations. In smooth regions, an efficient and accurate discretization is achieved via local p-adaptation. At strong discontinuities and shocks, a finite volume scheme on an h-refined element-local subgrid gives robustness. Thus, we obtain a hp-adaptive scheme that exploits both the high convergence rate and efficiency of a p-adaptive high order scheme as well as the stable and accurate shock capturing abilities of a low order finite volume scheme, but avoids the inherent resolution loss through h-refinement. A single a priori indicator, based on the modal decay of the local polynomial solution representation, is used to distinguish between discontinuous and smooth regions and control the p-refinement. Our method is implemented as an extension to the open source software FLEXI. Hence, the efficient implementation of the method for high performance computers was an important criterion during the development. The efficiency of our adaptive scheme is demonstrated for a variety of test cases, where results are compared against non adaptive simulations. Our findings suggest that the proposed adaptive method produces comparable or even better results with significantly less computational costs.

We present an efficient hp-adaptive discretization for sharp interface simulations of compressible twophase flows using the level-set ghost fluid method. The discretization employs a high order p-adaptive Discontinuous Galerkin (DG) scheme in regions of high regularity, whereas discontinuities are captured by a more robust Finite Volume (FV) scheme on an element-local sub-grid. The h-refinement strategy effectively carries over the subscale resolution capability of the DG scheme to shocks and the phase interface, while preserving an essentially non-oscillatory behavior of the solution. The p-refinement and the FV-limiting are controlled by a common indicator that evaluates the modal decay of the solution polynomials. The resulting adaptive hybrid DG/FV operator is used for the governing equations of both, the fluid flow and the level-set transport. However, the hp-adaptive discretization, together with solving the computationally expensive levelset equations only in the vicinity of the phase interface, causes pronounced variations in the element costs throughout the domain. In parallel computations, these variations imply a significant workload imbalance among the processor units. To ensure parallel scalability, the proposed discretization thus needs to be complemented by a dynamic load balancing (DLB) approach. We introduce a DLB scheme that determines the current workload distribution accurately through element-local walltime measurements and repartitions the elements efficiently along a space-filling curve. We provide strong scaling results to underline the parallel efficiency of the presented hp-adaptive sharp interface framework. Moreover, complex benchmark problems demonstrate that it handles efficiently and accurately the inherent multiscale physics of compressible two-phase flows.

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