| Mmay interesting problems in physics mad engineering aJ'e modeled by dissipativedynmnical systems These systems possess a bounded absorbing set that all trajecto-ries enter in a finite time and thereafter remain inside,for example,the Navier-Stokesequations,the Lorenz equations and several other equations People hope numericalmethods can inherit the important property ofthe systems when used to solve these sys-tems The study of dissipativity still is the importmat object in the study of dynamicalsystems themself mad their numerical methodsThis paper is concerned with the dissipativity of system itself and the numericalmethods for nonlinear NDDES:whereÏ„1,Ï„2,is real positive constants Let H be a real or complex Hilbert space withthe inner product(·,·)mad the corresponding norm ll·ll,W be a dense continuouslyimbedded subspace of H,φ:(一∞,0]â†'w is a given continuous and differentiablefunction,f:[0,+∞)×W×W×Wâ†'H is a given locally Lipschitz continuousmapping satisfying the following conditions for all t≥0,y,u,v∈Wwhereα(t)is bounded and non-negative continuous functions in[0,+∞),β(f),o-(t),γ1(t),γ2(t),,Ly(f),Lu(t)and Lv(t)are bounded continuous functions in[0,+∞)In this paper,we obtain the main results as follows:(1)The dissipativity result ofthe above system itselfis givenï¼›(2)It is proved that a DJ-irreducible and algebraically stable Runge-Kutta methodwith appropriate interpolation operator is dissipative for finite-dimensional dynamicalsystems in NDDEs We also prove that a(k,l)-algebraically stable Runge-Kutta meth-ods is dissipative for infinite-dimensional systems in NDDEsï¼›(3)A sufficient condition is given which makes G(c,p,O)-algebraically stable one-leg methods to keep the dissipativity ofthe original problem itselEThese results are more generally mad wider than the related existing results in theliteratures Finally the numerical example further demonstrate the validity ofthe theo-retical analysis... |
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