A Forward-Backward Time Integration Scheme to Treat Internal Gravity Waves

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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 he avoided by applying the trapezoidal rule to calculate the pressure field from the hydrostatic equation.

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