On the problem of optimal approximation of the four-wave kineticintegral amg

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