On the problem of optimal approximation of the four-wave kinetic integral

Abstract

Abstract. The problem of optimization of analytical and numerical approximations of Hasselmann's nonlinear kinetic integral is discussed in general form. Considering the general expression for the kinetic integral, a principle to obtain the optimal approximation is formulated. From this consideration it follows that the most well-accepted approximations, such as Discrete Interaction Approximation (DIA) (Hasselmann et al., 1985), Reduced Integration Approximation (RIA) (Lin and Perry, 1999), and the Diffusion Approximation proposed recently in Zakharov and Pushkarev (1999) (ZPA), have the same roots. The only difference among them is, essentially, the choice of the 4-wave configuration for the interacting waves. To evaluate a quality of any approximation for the 2-D nonlinear energy transfer, a mathematical measure of relative error is constructed and the meaning of approximation efficiency is postulated. By the use of these features it is shown that DIA has better accuracy and efficiency than ZPA. Following to the general idea of optimal approximation and by using the measures introduced, more efficient and faster versions of DIA are proposed.

Swell and Slanting-Fetch Effects on Wind Wave Growth

“…However, it is yet unclear that the increase in computational effort is warranted given the uncertainty in the other source terms. A balance must be found between the cost of the computation and its accuracy (e.g., Polnikov and Farina 2002).…”

Wave modelling – The state of the art

“…Still, a number of attempts have been made to intercompare different computational methods. Lavrenov (2001 (Polnikov and Farina, 2002;Polnikov, 2003). Using a certain definition of the error measure, it was shown that the DIA approximation is the best one among some other theoretical approximations: the diffusion approximation (Zakharov and Pushkarev, 1999) and the reduced integration approximation (Lin and Perrie, 1999).…”

“…Komatsu (1996) (referred to Hashimoto et al, 2002) developed the SRIAM, which is a multiple DIA based on the exact RIAM method. Polnikov and Farina (2002) proposed a version of the fast DIA, which doubles the speed of calculations without loss of accuracy.…”

The choice of optimal Discrete Interaction Approximation to the kinetic integral for ocean waves

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Abstract.A lot of discrete configurations for the fourwave nonlinear interaction processes have been calculated and tested by the method proposed earlier in the frame of the concept of Fast Discrete Interaction Approximation to the Hasselmann's kinetic integral (Polnikov and Farina, 2002). It was found that there are several simple configurations, which are more efficient than the one proposed originally in Hasselmann et al (1985).

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“…At present, this point is very well elaborated, and the only problem is to find the optimal approximation to the kinetic integral, which has the best balance of accuracy and speed of calculations. This problem was considered in detail in our previous paper (Polnikov and Farina, 2002), where one can find a good list of proper references. 1 On the basis of specially constructed mathematical measures and definitions (which is called PF-methodic), the authors of PF have shown that among different modern approximations of the kinetic integral the best one is the so-called Discrete Interaction Approximation (DIA), proposed in Hasselmann et al (1985).…”