Computer Algebra Recipes

7.3. SIMULATING SOLITON COLLISIONS 311
An operator G is introduced to calculate the rhs of the Zabusky{Kruskal algorithm
(7.6) (with à replaced with U) for a speci¯ed i and j.
> G:=(i,j)->U(i,j-1)-r*h^2*(U(i+1,j)+U(i,j)+U(i-1,j))
*(U(i+1,j)-U(i-1,j))/3
-r*(U(i+2,j)-2*U(i+1,j)+2*U(i-1,j)-U(i-2,j)):
The numerical algorithm is ¯rst iterated from i =2toM ¡ 2foragivenjvalue and then from j =1toN. The spatial index i isstartedat2andendedat
M ¡ 2 to avoid unknown U values from the two edges of the grid entering into
the double do loop calculation.
> for j from 1 to N do;
> for i from 2 to M-2 do
> U(i,j+1):=G(i,j):
> end do:
> end do:
A graphing operator is formed to plot the entire pro¯le on the jth time step.
> gr:=j->plot([seq([i,U(i,j)],i=2..M-2)],thickness=2):
Using every second graph, the sequence of pictures is now animated.
> plots[display]([seq(gr(2*j),j=0..N/2)],insequence=true);
You will have to execute the recipe, click on the resulting plot, and then on the
start arrow to see the wonderful animation. Despite the initial overlap of the
solitary-wave tails and the coarse spatial grid used (recall, h = 1), the solitary
waves are remarkably stable, all three surviving the collision process apparently
unchanged. After the collision the order of the waves is the reverse of the initial
ordering, with the smallest pulse on the left and the largest on the right.
> cpu:=time()-begin;
cpu := 29:072
The CPU time on a 3-GHz PC is about 29 seconds.
Although we have concentrated on soliton collisions here, the recipe may
be easily modi¯ed to investigate the behavior of other input pro¯les. For most
cases, an analytic solution will not exist and the numerical simulation route is
the only feasible one to take.
PROBLEMS:
Problem 7-17: Third derivative
Using the Taylor expansion, derive the approximation (7.5) to @3Ã=@3x. Problem 7-18: A di®erent scheme
In the Zabusky{Kruskal ¯nite di®erence scheme for the KdV equation, Ã in the
nonlinear term à (@Ã=@x) was approximated by the average of three à terms
at the grid points (i +1;j), (i; j), and (i ¡ 1;j). Compare the results obtained
in the text recipe with those you would obtain if U ´ Ã were approximated by
Ui;j alone. Discuss your result.