A bound- and positivity-preserving path-conservative discontinuous Galerkin method for compressible two-medium flows

Abstract

This work presents a high-order path-conservative Runge-Kutta discontinuous Galerkin method to simulate compressible two-medium flows by solving a γ-based model with the stiffened equation of state. The main contributions are as follows. Firstly, the path-conservative discontinuous Galerkin method is used to solve the γ-based model and is able to preserve uniform velocity and pressure fields around an isolated material interface. Secondly, a conservative-variables-based affine-invariant weighted essentially non-oscillatory limiter is employed to suppress nonlinear instability in the vicinity of discontinuities. Furthermore, an adaptive local Lax-Friedrichs numerical flux is adopted to improve the numerical resolutions. Last but not least, a bound- and positivity-preserving limiting strategy with strict theoretical analysis is developed for the stiffened equation of state to avoid the occurrence of inadmissible solutions while improving the robustness of the simulations. The h-adaptive Cartesian mesh is used for the numerical experiments to verify the validity of proposed method, and the numerical results of various one- and two-dimensional benchmark test cases demonstrate that the proposed method can efficiently and accurately handle complex two-phase flow problems involving pronounced interface deformations and large pressure ratios up to 10⁵.