A method for construction of second-order accuracy difference schemes permitting no false two-grid-interval wave in the height field

Abstract

A method has been developed to inherently prevent the false two-grid-interval wave in height field, which is, to the author's knowledge, at present permitted as a stationary solution to the gravity-wave part of all space-centered schemes used to solve the primitive equations in numerical forecasting and simulation experiments. The proposed method preserves space- and time-centered differencing and second-order accuracy. It involves introduction of auxiliary velocity points midway between the neighboring height grid points. Velocity components at these auxiliary points are assumed to have space-averaged values at the beginning of a considered time step; acceleration contributions are evaluated and added to these initial values to obtain components at the middle or at the end of the time step. Resulting velocity components are then used for a more accurate calculation of the divergence term in the continuity equation. The procedure leads to schemes which propagate single-grid-point height perturbations to all the existing height grid points, and thus do not permit formation of a false high wave number noise in the forecast fields. For the Heun, leapfrog, and implicit time differencing schemes an analysis is performed of the effect that this procedure has on the properties of the scheme. For the Heun scheme, the procedure results in a reduction of the instability of the scheme, and in a reduction of the truncation error. For the leapfrog scheme slight instability is produced; however, values of the amplification factors, and a numerical example, indicate that this modified leapfrog scheme can still successfully be used for integration of the primitive equations. For the implicit scheme, the procedure preserves the unconditional stability of the scheme, and also reduces the truncation error. Finally, a numerical example is shown, giving a dramatic illustration of the advantages of the proposed method.DOI: 10.1111/j.2153-3490.1973.tb00629.x