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 * Corresponding author Email address: tian_baolin@iapcm.ac.cn (Baolin Tian)