A second-order accurate difference scheme is developed to study cavitation in unsteady, one-dimensional, inviscid, compressible flows of water with gas. As an extended version of the standard MUSCL scheme, this scheme is based on the solutions of local gas-water-vacuum initial value problems. Numerical results are presented for gas-water Riemann problems to demonstrate the performance of the scheme. The scheme is also applied to simulate the cavitation process of the flow in a water shock tube.