A bound-preserving oscillation-eliminating discontinuous Galerkin method with operator splitting for solving Kapila's five-equation model

Abstract

This paper proposes a robust operator-splitting discontinuous Galerkin (DG) framework to overcome the severe stiffness-induced instabilities in simulating compressible two-phase flows governed by Kapila's five-equation model with the Tammann equation of state. Specifically, the system is decoupled into a five-equation transport model and a stiff $\hat{I}º$-source term. The former is discretized via a quasi-conservative DG method \cite{cheng2020quasi}, while the latter is resolved by the local DG method combined with a novel adaptive implicit strategy that hybridizes the backward Euler and second-order singly diagonally implicit Runge-Kutta schemes. This implicit strategy possesses the unconditionally bound-preserving property, and thus effectively circumvents the severe stability constraints and time-step penalties inherent in traditional explicit schemes. Furthermore, to enhance computational robustness, we integrate an oscillation-eliminating DG (OEDG) procedure to suppresses spurious oscillations without characteristic decomposition, complemented by a bound-preserving limiter to maintain physically admissible numerical solutions. We also prove that the proposed operator-splitting DG framework, integrated with the oscillation-eliminating limiter, and the bound-preserving limiter, strictly satisfies the Abgrall condition. Finally, extensive numerical experiments are conducted to demonstrate the superior robustness and efficiency of the method.