C oordinate T ran sfo rm atio n

T h a t m eans in th e transform ed plane ( X , P ) th e initial condition of (5.35a) is sim ply

P { X , - l ) = f { X ) = - X . (5.37)

In fact, th e idea of choosing —f~^ as th e transform ation function is generated from th e tre a tm e n t of nonzero ji later. In th a t case P and th e Cauchy integrals are found to exhibit little change outside an X range of 0 (1 ) near th e origin w ith tim e m arching before a singularity or instability is reached, and th e corresponding finite-part integrals(F P I) vary dram atically around th e origin. So we need a tran sform atio n to cluster grid points around th e origin. F ig u r e 5.1 shows th e d istrib u tio n of th e grid in th e (%, /)-p la n e , which corresponds to equal spacing of X . We see th a t this transform ation does cluster grid points tow ards th e origin. Obviously, initially equal spacing in th e transform ed plane ( X , P ) is optim al since P { X j —1) = — so is th e corresponding non-equal spacing in th e physical plane (X , P ). If th e solution changes little w ith tim e m arching, b o th th e equal spacing in th e transform ed plane and th e non-equal spacing in th e physical plane can be expected to be approxim ately optim al. However, for th e p, = 0 case, solutions can be expected to develop traveling waves which vary over a relatively large spatial range as tim e is m arched forward. R esults later confirm this. In consequence th e above choice of th e transform ation function m ight not be a very good one then. It is found however th a t th e transform ation in fact also works very well here due to th e stretching effect, which can prevent th e propagation of errors from any ou ter boundary.

Y=fXX) 2

2

X

Figure 5.1: The Distribution of Grids. Yk = f{Xk), Xk = g{Xk) = - f ^{Xk), Xk - 5 + kh, k = 0, l , 2, -- - , N . h = 10/#, N = 80.

Based on the above choice, we also have

g~^{Yoo) = Too, g~^{ — oo) = —oo. (5.38)

Hence (5.35b) and (5.35c) can be fixed. The boundary conditions of (5.35) now stay intact, i.e. (5.2), and the initial condition is (5.37).

Therefore we now have two com putational models, one being (5.5), (5.6) and (5.2) in the physical plane and the other being (5.35), (5.37) and (5.2) in the transform ed plane. In the following discussion we discretize both models and then perform some comparisons.

5.4

T h e T reatm ent o f C auchy In tegrals and B a ­

sic Idea o f th e A lg o rith m s for N o n zero

C ase

The nature of (5.5) with nonzero /t or 05 0 is very much different from th a t of the standard KdV equations we discussed in chapter 3, due to the extra

integral term s. Consequently th e algorithm s can be expected to be m uch m ore com plicated now. In order to discretize (5.5) in a finite integration interval, here, we need to m anipulate (5.5) further.

F irst, we m ay rew rite (5.5) as follows

A, CLt\ Pt -\-PPx — ^ ^ P x x x + j ^ P x { I H— { J+«/i2), (5.39a) 7T where

/ ' -

P

+ PP

j g

(5.39b)

£

X - S

(5.39c)

are th e leading contributors to th e corresponding Cauchy integrals; while

J - o . P { X , T ) - P { S , T ) J x , P ( X , T ) - P { S , T )

(5.39d) and

+ (5.3 9c)

are th e corresponding Cauchy integral residuals whose effect on the whole in te­ grals is expected to be negligible as long as th e integration interval [Xi,Xr] is large enough to cover th e m ain influential range of th e integrals. Therefore we th en drop th e residual term s to give

Pt + P P x = claPx x x + P^PxI H— - J . (5.40)

7T

So th e discretization to be processed is essentially applied to (5.40) as an ap­ proxim ation of (5.5), w ith th e finite p a rt integrals (F P I) (5.39b) and (5.39c) as approxim ations of the Cauchy integrals (5.9) and (5.11). In practice, this kind of approxim ation can be justified by exam ining th e effect of enlarging th e in teg ra­ tion intervals or by introducing th e far-field asym ptotic evaluation of th e Cauchy integral residuals.

C orrespondingly, in the transform ed plane we have th e following counterparts of (5.40), (5.39b) and (5.39c) respectively.

Pt P F — (%4

P x x 9 Px

_{+ iiF i + — j ,}

_(5.41a)

Furtherm ore, we notice th a t th e finite p a rt integrals (5.39b) or (5.41b) can be w ritten in an im plicit form w ith respect to / or

7

' ^ ' a ^ P s s s { S , T ) + i i P s i S , T ) I ( X , T ) + f J ( S , T ) I { X , T ) = - f Jx

P i X , T ) ^ F ( S , T )

where E { X , T ) = F ( X , T ) / g \ X ) . T h at is, it is possible to elim inate th e deriva­ tives of P or P w ith respect to T in th e expressions of th e finite p a rt integrals.

In th e forthcom ing discussion, we first derive several form ulae to handle th e finite p art integrals (5.42a) and (5.39c) as well as (5.42b) and (5.41c), and then propose several possible algorithm s for the system (5.40), (5.39c) and (5.42a).

5 .4 .1

F in ite P a r t In te g r a ls

O ur basic idea for th e discretization of finite p a rt integrals is to use Taylor expansions. Now we illu strate this idea in th e physical plane.

For simplicity, we om it T in th e expression of (5.42a), giving

I { X ) = £ ' (5.43a)

where

7T

(5.43c)

I P W ) 'f 5 = X .

W riting (5.43a) in the form of sums in th e subintervals [Xj_i j Xj_^i],j = 1, 2, - - , N — I, gives