Entropy correction analyses for Roe scheme

Three types' entropy fix formulations for the Roe scheme were analyzed in theory and assessed by three numerical tests which were shock tube problem, step flow and double Mach reflection. The results show that "Carbuncle phenomena" occurs when capturing strong shock because of the original Roe scheme's numerical instability. Muller's and Harten-Yee's entropy fix methods introduce numerical dissipation to both shock and non-physical expansion shock processes. The numerical dissipation improves the Roe scheme's numerical stability and cures "Carbuncle phenomena" entirely for shock case and diffuses the expansion shock completely as well. Harten-Hyman type's entropy correction formulations do no contribution to shock case and can not cure the "Carbuncle phenomena". They can only fix non-physical expansion processes partially because their numerical dissipation is not large enough. The eigenvalue correction method which directly uses delta replace eigenvalue itself exceeds the traditional method.