Secondary conservation properties of compact schemes in split forms

This study theoretically proves that the properties of kinetic energy preservation and product rule discretely hold in a non-dissipative numerical scheme based on a compact scheme. A previous work developed a kinetic energy and entropy preserving (KEEP) compact scheme and showed superior numerical stability compared to the standard compact scheme. However, the reason for the improved numerical stability in the KEEP-compact scheme was not clearly discussed in the previous work. This study reviews the construction of the compact scheme using the Padé approximation and shows that the important discrete properties of kinetic energy preservation and product rule, which play major roles in improving numerical stability in the existing explicit KEEP schemes, hold in the KEEP-compact scheme.