NonlinearSchrodingerequation(NSE)arisesinmanyphysicalproblems.Itisaveryimportantequation.Alotofworksstudiedthewellposed,theexistenceofsolutionofNSEetc.Andtherearemanyworksstudiedthenumericalmethodsforit.Recently,sincethedevelopmentofinfinitedimensionaldynamicsystemthedynamicalbehaviorofNSEhasbeeninvestigated.Thepaper[1]studiedthelongtimewellposedness,theexistenceofuniversalattractorandtheestimateofLyapunovexponentforNSEwithweaklydamped.Atthesametimeitwasneedtostudythelargetimenewcomputationalmethodsandtodiscussitsconvergenceerrorestimate,theexistenceofapproximateattractorsetc.InthispapewestudytheNSEwithweaklydamped(1.1).Weassume,where0<λ<2isaconstant.Ifwewishtoconstructthehigheraccuracycomputationalscheme,itwillbedifficultthatstaighfromtheequation(1.1).Thereforewestartwith(1.4)andusefullydiscreteFourierspectralmethodwithtimedifferencetodisscus
