Numerical recipes

856 Chapter 19. Partial Differential Equations
(19.3.13). Called the alternating-direction implicit method (ADI), this embodies the
powerful concept of operator splitting or time splitting, about which we will say
more below. Here, the idea is to divide each timestep into two steps of size ∆t/2.
In each substep, a different dimension is treated implicitly:
u n+1/2
j,l
u n+1
j,l
= u n j,l
= un+1/2
j,l
1
+
2 α
�
+ 1
2 α
δ 2 x un+1/2
j,l
�
δ 2 x un+1/2
j,l
+ δ 2 y un j,l
�
+ δ 2 y un+1
j,l
�
(19.3.16)
The advantage of this method is that each substep requires only the solution of a
simple tridiagonal system.
Operator Splitting Methods Generally
The basic idea of operator splitting, which is also called time splitting or the
method of fractional steps, is this: Suppose you have an initial value equation of
the form
∂u
∂t
= Lu (19.3.17)
where L is some operator. While L is not necessarily linear, suppose that it can at
least be written as a linear sum of m pieces, which act additively on u,
Lu = L1u + L2u + ···+ Lmu (19.3.18)
Finally, suppose that for each of the pieces, you already know a differencing scheme
for updating the variable u from timestep n to timestep n +1, valid if that piece
of the operator were the only one on the right-hand side. We will write these
updatings symbolically as
u n+1 = U1(u n , ∆t)
u n+1 = U2(u n , ∆t)
···
u n+1 = Um(u n , ∆t)
(19.3.19)
Now, one form of operator splitting would be to get from n to n +1by the
following sequence of updatings:
u n+(1/m) = U1(u n , ∆t)
u n+(2/m) = U2(u n+(1/m) , ∆t)
···
u n+1 = Um(u n+(m−1)/m , ∆t)
(19.3.20)
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