A Conservative Six-Equation Multi-material Method for Compact Finite Differences

High-order compact finite difference schemes' spectral-like accuracy is well suited for capturing turbulent mixing, but additional stabilization is needed to capture jumps in material properties in multi-material flows. Localized artificial diffusivity (LAD) methods have been developed to stabilize large material property jumps while maintaining the high-order accuracy of compact finite difference methods, without the need for limiters or bounds guarantees. However, compact finite difference methods traditionally struggle for materials with disparate equations of state, because the P/T equilibrium is enforced with mixture equations of state, which fail to converge. To address this limitation, we present a six-equation model for the compact finite difference method. This method relaxes the pressure equilibrium requirement by transporting separate material energies and using a mixed zone closure law based on the materials' bulk moduli. The model is stabilized with only artificial diffusivities without limiters or bounds guarantees. The method is formulated to conserve mass, momentum, and energy. The strength of the proposed scheme is validated on a suite of test problems.