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75) is then d J(u, v) = 0. 76), the modiﬁed local energy becomes v2 (ux )2 G(u, ux , v) = + . 91). 32 Discrete Variational Derivative Method [step 1d ] Deﬁning a discrete energy We deﬁne a discrete modiﬁed local energy Gd , again replacing u, v with (m) Uk (m) , Vk , and ux with some diﬀerences. 95) and accordingly a global energy as N d Jd (U (m) , V (m) ) ≡ ′′ Gd,k (U (m) , V (m) )∆x. 97) in a discrete setting. 98) to ﬁnd the discrete variational derivatives, δ Gd δ(U (m+1) ,U (m) )k and δ Gd δ(V (m+1) , V (m) )k .
34) are used. 57a) 〈2〉 Uk (m+1) + Uk (m) 2 . 47). 47)) With given initial data U (0) and appropriate boundary conditions, we compute U (m) by, for m = 0, 1, 2, . , i Uk (m+1) − Uk (m) ∆t =− δGd δ(U (m+1) , U (m) )k , k = 0, . . , N. 58) Introduction and Summary 23 The scheme automatically becomes conservative as follows. 4 is conservative in the sense that Jd (U (m) ) = Jd (U (0) ), PROOF m = 1, 2, 3 . . ) = 0. 58) is used. 57b). Again this is just the standard Crank–Nicolson scheme. 51). In the subsequent chapters, we will deal with the following complex-valued PDEs.
To this end, let us deﬁne a discrete local energy by G(u, ux ) = d Gd,k (U (m) ) ≡ 1 2 (δk+ Uk (m) )2 + (δk− Uk (m) )2 2 + (Uk (m) )4 . 124) which is dissipative in the sense that N N ′′ k=0 Gd,k (U (m+1) )∆x ≤ ′′ Gd,k (U (m) )∆x, m = 0, 1, 2, . . 125) k=0 Now observe carefully how nonlinearity is “passed down” from the energy function G(u, ux ) to the resulting PDE, and at the same time, the discrete energy function Gd to the scheme. 23) is deﬁned with the variational derivative of the energy G.