A bound-preserving oscillation-eliminating discontinuous Galerkin scheme for compressible two-phase flow

TL;DR

This paper introduces a high-order, bound-preserving discontinuous Galerkin scheme with an innovative operator-splitting strategy for stable, accurate simulation of two-phase flows governed by complex models.

Contribution

It develops a novel operator-splitting approach with unconditionally bound-preserving implicit treatment for stiff source terms in two-phase flow simulations.

Findings

The implicit scheme is proven unconditionally bound-preserving.

The method effectively handles stiff source terms without stability issues.

Numerical tests show superior robustness and efficiency in complex flow scenarios.

Abstract

This paper presents a high-order bound-preserving oscillation-eliminating discontinuous Galerkin (BP-OEDG) scheme for simulating gas-gas and gas-liquid two-phase flows governed by the Kapila five-equation model with the Tammann equation of state (EOS). The primary computational bottleneck arises from the severe CFL restriction imposed by the stiff $\kappa$-source term in the volume fraction equation. To circumvent this, we propose a novel operator-splitting strategy that decouples the system into a transport model and a stiff $\kappa$-source term. The former is discretized via a quasi-conservative DG method \cite{cheng2020quasi}, while the latter is resolved by an adaptive implicit strategy hybridizing the backward Euler and SDIRK2 methods. We rigorously prove that this implicit treatment is unconditionally BP, effectively removing the stiffness-induced stability constraints inherent in traditional explicit schemes. To further enhance precision, a velocity divergence reconstruction inspired by the Local Discontinuous Galerkin (LDG) method is integrated into the implicit solver. Furthermore, an OE limiter is employed to suppress spurious oscillations without characteristic decomposition, complemented by a BP limiter to ensure the BP property of partial densities, pressure, and volume fraction. Crucially, we prove that the proposed BP-OEDG scheme, integrated with the splitting strategy, strictly satisfies the Abgrall condition. Extensive numerical experiments, including challenging water-air shock-bubble interactions, demonstrate the superior robustness and efficiency of the method.