Computer Algebra Recipes

6.3. NUMERICAL SIMULATION OF PDES 279
ical algorithm for the KGE then is
(Ãi+1;j ¡ 2 Ãi;j + Ãi¡1;j)
h 2
¡ (Ãi;j+1 ¡ 2 Ãi;j + Ã i;j¡1)
k 2 = aÃi;j; (6.22)
or, setting r ´ k 2 =h 2 and c ´ 2 ¡ 2 r ¡ k 2 a, and rearranging,
Ãi;j+1 = rÃi¡1;j + cÃi;j + rÃi+1;j ¡ Ãi;j¡1; (6.23)
with j = 1; 2; 3; :::: The mesh points involved in this explicit scheme are
schematically depicted in Figure 6.3.
ψ i,j+1
ψ ψ ψ
i−1,j i,j i+1,j
ψ i,j−1
Figure 6.3: Relevant mesh points for numerically solving the KGE.
In this algorithm, the unknown à value on time step (j +1) depends on
its values on the previous two time steps, j and (j ¡ 1). The second time
row, corresponding to taking j = 1, will be the ¯rst to be calculated. The Ã
values are known along the zeroth time row from the initial condition Ã(xi; 0) =
f(xi). To apply the scheme, the à values along the ¯rst time row must also be
known. These may be determined from the initial transverse velocity. Using
the forward-di®erence approximation, the condition _ Ã(x; 0) = g(x) yields 6
Ãi;1 ¡ Ãi;0
= g(xi);
k
or Ãi;1 ´ G = Ãi;0 + kg(xi): (6.24)
Now Russell programs the explicit scheme (6.23), ¯rst loading the Linear-
Algebra package so a matrix approach can be used again.
> restart: with(LinearAlgebra): begin:=time():
6 In the problem set, Russell will show us a better approximation than this one.