The method of characteristics with fixed-grid intervals used to predict pipeline transients can be enhanced by projecting characteristics from outside the fundamental grid size. At interior regions linear interpolations are used which satisfy the required stability criterion and at boundaries linear interpolations along time lines complete the numerical solution. An error analysis shows the effects of interpolation, spacing, and grid size on numerical attenuation and dispersion; the results suggest ways to improve the predicted transients by adjusting three finite differencing parameters. Numerical solutions for constant and variable wave speed conditions are presented and show how high-frequency components become less severely suppressed.