Several computational methods that are designed to resolve a contact surface more sharply than that given by conventional upwind schemes are explored. Two different numerical methods are newly developed where one approach employs an overset mesh arrangement to follow the material interface precisely, and the other one considers two component gases to define the material interface and try to keep it discontinuous by modifying the numerical flux function. Details of the formulation are described for the latter scheme and typical numerical results are presented. The calculated results for Richtmyer-Meshkov instability problem using these schemes are compared with that given by the Piecewise Parabolic Method (PPM) scheme. The growth rate of the instability of the material interface for linear regime is compared. From the results, we emphasize that the present new scheme assuming two component gases has a unique capability of resolving a contact surface quite sharply and accurately.