Two finite-difference methods for geophysical fluid problems are described, and stability conditions of these schemes are discussed. These two schemes are formulated based upon a similar procedure given by Lax and Wendroff in order to obtain a second-order accuracy in finite-difference equations. However, the two schemes show remarkable differences in their computational stability. One scheme is stable, as one might expect, under the usual stability conditions of Courant-Friedrichs-Lewy and Lax-Wendroff. However, the other scheme is conditionally stable only if the flow is supereritical (supersonic in the case of gas dynamics) and unconditionally unstable if the flow is suberitical (subsonic).