Unformatted text preview: 3 (u
x y n x y t jk = txxyy ) +1 2 + : : : ]:
n jk = n y jk 5.2. Operator Splitting 9 Thus, the local discretization error of the ADI method is
n
jk =( )
n jk CN + O( t2) = O( x2) + O( y2) + O( t2) which is the same order as that of the CrankNicolson method.
The stability of (5.1.5) can be analyzed by the von Neumann method. The twodimensional form of the discrete Fourier series is U=
n ;1 ;1
XX
J jk p K =0 =0 A e2
n pq ( i pj=J + qk=K ): (5.1.8a) q Substituting into (5.1.5 and proceeding as in one dimension, we nd A = (M ) A0
n (5.1.8b) n pq pq pq where A0 is a Fourier component of the initial data and M is the ampli cation factor. Again, following the onedimensional analysis, we verify that jM j 1 for all
positve r and r hence, the PeacemanRachford version of the ADI method (5.1.5) is
unconditionally stable.
pq pq pq x y 5.2 Operator Splitting Methods
The ADI approach is often di cult to extend to problems on nonrectangular domains,
to nonlinear problems, and to problems having mixed derivatives such as u . The
dimensional reduction developed for the ADI method can be viewed as an approximate
factorization of the di erential or discrete operator. Let us motivate the factorization by
rst examining the ordinary di erential equation
xy dy = (a + b)y
dt
which, of course, has the solution y(t) = e ( + ) y(0) = e e y(0):
ta b ta tb The latter form suggests that the solution of the initial value problem may be obtained
by rst solving dy=dt = by to time t with y(0) prescribed as initial data, and then solving 10 MultiDimensional Parabolic Problemss dy=dt = ay subject to the initial condition e y(0). This interpretation, however, does
tb not extend to vector systems of the form dy = (A + B)y
dt unless A and B commute. Thus, we may write the solution of the vector problem as y(t) = e (A+B) y(0)
t where t2
e C = I + tC + 2! C2 + : : : :
t However, y(t) = et(A+B)y(0) = etAetBy(0)
6 unless AB = BA. Nevertheless, let's push on and consider a linear partial di erential
equation u = u = ( 1 + 2 )u
L t L (5.2.1) L where L is a spatial di erential operator that has been split into the sum of L1 and L2.
We'll think of L1 as being associated with x derivatives and L2 as being associated with
y derivatives, but this is not necessary. Any splitting will do.
The solution of the linear partial di erential equation can also be written as the
exponential
u(x y t) = e L u(x y 0)
t when L is independent of t. The interpretation of the exponential of the operator
follows from a Taylor's series expansion of u in powers of t, i.e., L 2
@
u(x y t) = u(x y 0) + tu (x y 0) + t u (x...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis

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