Economical time integration schemes, which preserve important meteorological waves fairly well, while damp effectively high-frequency noises, are proposed. In order to reduce computation times, low-frequency terms in the primitive equations, which need expensive times for computation, are assumed to be constant within a several time steps. Furthermore, the forward-backward scheme is applied to the integration of high-frequency terms. The integration is carried periodically with the combination of one calculation of obtaining a constant value of low-frequency terms and several forward-backward integrations of high-frequency terms.
The constant value used for low-frequency terms is calculated from the weighted mean of the actual value at a certain time level of integration and the temporarily integrated value from that time with a relatively long time step. Two different schemes are proposed to evaluate the temporarily integrated value. One, in which both the low-frequency and the high-frequency terms in the primitive equations are used in the temporary integration, is called Scheme 1 in this report, and the other, in which only the low-frequency terms are used, is called Scheme 2.
Moreover, at the backward step of the forward-backward technique a certain weight parameter is multiplied to high-frequency terms pre-predicted at the forward step. This technique is also introduced into the adjustment stage of the so-called Gadd's split explicit scheme. This revised Gadd scheme is called Scheme 3 in this report.
These three scheme are applied to the one-dimensional linearized shallow water equations and their stability properties are described as a function of a wave-frequency and a weight parameter. It is shown that all schemes are stable over a certain range of wavefrequency, if a weight parameter used at the backward step is larger than unity. It is alsoshown that a wave-amplitude suffers more damping for a larger value of a weight para meter, while the maximum allowable time increment decreases.
In order to demonstrate the usefulness of these schemes, they are applied to a 5-daysprediction of one dimensional linearized primitive equations, and the predicted results are compared with the analytic solution. The followings are concluded. First of all, all schemes reduce the computation time to about 1/5 of the time required by the iterative time integration scheme. Secondly, the meteorological waves are preserved almost perfectly, while the high-frequency gravity waves are damped effectively. Although all schemes give almost the same results, it seems that Scheme 1 gives the best agreement to the analytic solution of a meteorological wave.
In order to achieve more economy of computation, the hybrid method is proposed. In this method, the scheme using a large weight parameter with a short time increment and the scheme using a small weight parameter with a relatively long time increment are applied aternately. By applying this method, the computation time is reduced to about 1/10 of the original iterative time integration scheme.
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