The conservative or dissipative schemes presented in the previous chapters were all nonlinearly implicit with respect to the numerical solution at the next time step. Although they had their own beauty and superiority in that they keep the desired dissipation or conservation property, the nonlinearity becomes more di±cult as the sizes of the target problems increase. One way of surviving this is to look for a better nonlinear solver. In fact, recently many practical solutions for large scale nonlinear equations have been devised. See Chapter 7 on this topic. Another practical compromise for this di±culty is to consider linearly im-

plicit versions of the dissipative or conservative schemes. Linearly implicit schemes are still implicit but linear with respect to the numerical solution at the next time step (recall Table 1.1 in Chapter 1). Obviously they are far cheaper than the nonlinearly implicit schemes. However, they generally lose the desired dissipation or conservation property in strict sense of the word; rather they hold in some more generalized senses, as will be described below. We have already seen the basics of constructing linearly implicit schemes in

Chapter 1, but let us review them again with a new example. Let us consider the cubic nonlinear SchrÄodinger equation (NLS):

i @u

@t = ¡uxx ¡ °juj2u; ° 2 R; (6.1)

u(j)(t; 0) = u(j)(t; L); j = 0; 1; 2: (6.2)

As to the NLS, we have already presented a conservative scheme in Chapter 4, which was nonlinearly implicit with respect to Uk(m+1). In order to obtain a linearly implicit scheme, it is essential to understand

the mechanism of how the nonlinearity in the energy is passed down to the equation through the variation calculation. In the cubic NLS equation, the nonlinear term juj4 in the local energy:

G(u; ux) = ¡juxj2 + °2 juj 4 (6.3)

is the source of the nonlinear term juj2u in the resulting equation (6.1). In general, the power in the nonlinear term in the energy is always one higher than that in the resulting nonlinear term. Hence we easily come to the conclusion that if we want the resulting scheme to be linear, we must restrict our discrete energy function to be quadratic, at most. In the cubic NLS, for example, this can be accomplished by breaking down jUk(m)j4 to jUk(m+1)j2jUk(m)j2, with the aid of Uk(m) (i.e., we make the energy function multistep). Its discrete variation calculation becomes

jUk(m)j2 Ã Uk

!³ Uk

(m+1) ¡ Uk(m¡1) ´

+ jUk(m)j2 Ã Uk

!³ Uk

(m+1) ¡ Uk(m¡1) ´ : (6.4)

Now jUk(m)j2(Uk(m+1) + Uk(m¡1))=2, which is the approximation of juj2u, is successfully linear with regard to the unknown variable Uk(m+1). Including the other terms as well, we can now construct a linearly implicit

scheme for the cubic NLS equation as follows. We de¯ne a multistep discrete local energy by

Gd;k(U (m+1);U (m)) d´

¡ ° 2 jUk(m+1)j2jUk(m)j2; (6.5)

and accordingly the discrete global energy by

Jd(U (m+1);U (m)) d´

00Gd;k(U (m+1);U (m))¢x: (6.6)

Uk = UkmodN : (6.7)

Taking its variation we have NX k=0

00Gd;k(U (m+1);U (m))¡Gd;k(U (m);U (m¡1))¢x =


±(U (m+1);U (m);U (m¡1))k

+ ±Gd

±(U (m+1);U (m);U (m¡1))k

2 ; (6.8)

where ±Gd

±(U (m+1);U (m);U (m¡1))k

= ¡1 2 ± h2i k (Uk

(m+1) + Uk(m¡1))¡ °2 jUk (m)j2(Uk(m+1) + Uk(m¡1)); (6.9)


±(U (m+1);U (m);U (m¡1))k =


±(U (m+1);U (m);U (m¡1))k (6.10)

are \three points discrete variational derivatives," which have been already introduced in Section 3.5. With them we can now de¯ne a linearly implicit ¯nite di®erence scheme as


à Uk

(m+1) ¡ Uk(m¡1) 2¢t