Consistent implementation of characteristic flux-split based finite difference method for compressible multi-material flows

TL;DR

This paper develops a new consistent high-order characteristic flux-split finite difference method for compressible multi-material flows, effectively handling interface discontinuities without special treatments and maintaining physical equilibria.

Contribution

A novel general numerical algorithm for high-order CFS-FDM that avoids special treatments and ensures consistency in simulating compressible multi-material flows.

Findings

Maintains velocity, pressure, and temperature equilibrium.

Handles interface interactions with shocks and rarefactions.

Eliminates need for special algorithmic treatments.

Abstract

In order to prevent velocity, pressure, and temperature spikes at material discontinuities occurring when the interface-capturing schemes inconsistently simulate compressible multi-material flows(when the specific heats ratio is variable),various non-conservative or quasi-conservative numerical models have been proposed. However, designing a consistent numerical algorithm, especially using the high-order characteristic flux-split based finite-difference method (CFS-FDM) is still an open question. In this study, a systematical analysis of previous algorithms of the consistent implementing the high-order CFS-FDM for such flows is performed, and the reasons of special treatments in these algorithms are revealed. Based on this analysis, a new general numerical methodology that successfully avoids any special treatments as those required in previously reported algorithms, is derived. In this new algorithm, we rewrite the non-conservative term as a conservative term with a source term containing velocity divergence. By consistently treating the advection velocity in the conservative term and velocity divergence in the source term by imposing a new additional criterion, specifically, that a multi-fluid algorithm should have the ability of maintaining a pure single-fluid, we finally derive a new general algorithm that does not need any special treatment, and is very convenient to implement. The results of some benchmark tests show that the final algorithm not only maintains the velocity, pressure, and temperature equlibria, but is also suitable for problems regarding the interaction of interfaces and strong shock and rarefaction waves.