Sun, W. Y., 1980: A forward-backward time integration scheme to treat internal gravity waves. Monthly Weather Review, 108 (4), 402-407.

Abstract: A forward time difference in the momentum equations and a backward time difference in the thermodynamic equation is conditionally stable according to von Neumann's stability analysis of linearized anelastic equations. In an anelastic-hydrostatic system, this scheme is also conditionally stable in a staggered coordinate, but it is unstable for a two-grid interval wave in a non-staggered coordinate. This unstable two-grid-interval wave can be avoided by applying the trapezoidal rule to calculate the pressure field from the hydrostatic equation.