High-order positivity-preserving method in the flux reconstruction framework for the simulation of two-medium flow

In this article, an accurate and robust numerical method is developed for solving the compressible two-medium model. The method in arbitrary high-order case has the following significant features: Firstly, it uses the discrete equations in the differential framework of flux reconstruction (FR). This simple discretization automatically satisfies the Abgrall condition and avoids the production of oscillations around isolated interfaces. Secondly, it solves the diffuse interface five-equation system coupled with the equation of state (EOS) for stiffened gases. The method is applicable to more fluid mediums while not relying on special treatments of material interfaces. Thirdly, it provides physically meaningful solutions that are bound-preserving for volume fractions and positivity-preserving for density, energy, etc. The robustness of the method is greatly enhanced owing to the maintenance of the hyperbolic characteristic of the system in various problems. Finally, it may couple with the weighted essentially nonoscillatory (WENO) limiter that we design for the five-equation model. The smoothness and monotonicity of the final results can be further improved. The above features are well examined and reflected in the relevant numerical experiments.