Computer Algebra Recipes

7.3. SIMULATING SOLITON COLLISIONS 313
then equation (7.11) reduces to
μ
dy
a dp + cdq+ edy =0: (7.13)
dx
For the SGE, a =1,b =0,andc = ¡1, so that (7.12) yields (dy=dx) 2 ¡ 1=0,
or dy=dx = §1. The two characteristic directions have slopes of 45 ± and ¡45 ± ,
respectively. Forming a diamond-shaped mesh with these slopes produces the
grid illustrated in Figure 7.9. Given the new grid, how is U (or Ã) calculated?
j=2
y
j=1
P
L R
(0,2) (2,2) (4,2)
(1,1) (3,1) (5,1)
j=0
(2,0) (4,0) (6,0)
i=0 i=1 i=2
Δx=h
x x
min max
Δy=h
Figure 7.9: Characteristic directions and labels for solving the SGE.
Consider the mesh point P in Figure 7.9, where it is desired to calculate the
unknown UP from the known values UL and UR on the previous time step. The
subscripts L and R denote advancing \from the left" along the characteristic
direction dy=dx = 1 and \from the right" along the characteristic direction
dy=dx = ¡1, respectively. Taking dy=dx = §1 and replacing (7.13) with a
¯nite di®erence approximation yields the following pair of equations,
(pP ¡ pL) ¡ (qP ¡ qL) =¡eL (yP ¡ yL);
¡(pP ¡ pR) ¡ (qP ¡ qR) =¡eR (yP ¡ yR);
x
(7.14)