A grid-free scheme for solving quasi-one-dimensional, isentropic, compressible flow in the subsonic regime is developed with a view toward its eventual generalization to three-dimensional turbulent compressible flow. The computational elements contain information about the dilatation and temperature fields. Velocity is recovered by summing over the contributions of individual elements. Differentiated terms in the governing equations are evaluated using a moving least-square fit with Gaussian weighting function. The transient flow in a suddenly constricted duct is computed including relaxation to an equilibrium solution. An accurate scheme to accommodate the passage of dilatation through the inflow and outflow boundaries is developed using the wave properties of the governing equations. The predicted duct flow matches the exact equilibrium solution and the expected properties of the transient wave field.