A numerical method for solving problems in which a moving surface of discontinuity separates regions of incompressible flow is presented. The method developed is notable in that it does not introduce any artificial smoothing of the change in fluid properties across the surface of discontinuity. This results in an increase in accuracy relative to methods which introduce smoothing effects. The method was also shown to be fairly versatile; problems describing a free surface, an immiscible fluid interface, and a premixed flame discontinuity were solved. There is a limitation, however, in that the method appears to be most suitable for application to inviscid problems. The reason for this limitation and possible approaches toward resolving it are discussed.