Вычислительная математика - ИСЭМ СО РАН

Forh ∈ (0, h 0 ], letx k = a + kh. A sequence of numerical values y k which approximatesy(x k ) can be
obtained by using the formula that Shampine and Watts [6] refer to which is given by Clippinger and
Dimsdale as,
y n+1 − 1 2 y n+2 = 1 2 y n + h 4 f n − h 4 f n+2
y n+2 = y n + h 3 f n + 4h 3 f n+1 + h 3 f n+2
(3)
Using vector notation and the matricesA = (a ij ),B = (b ij ), e = (1, ..., 1) T , and column vectors d =
(d 1 , d 2 , ..., d r ) T , y m = (y n+1 , y n+2 , ..., y n+r ) T , and F (y m ) = (f n+1 , f n+2 , ..., f n+r ) T . Equation 2.3 can
be written as
Ay m = hBF (y m ) + ey n + hdf n (4)
or
y m = hCF (y m ) + ey n + hdf n (5)
( 2
) ( ) (
−1
where B = 3 12
1 5
)
4 1 and e = , d = 12
3 3
1
1 .
3
For the numerical solutions by predictor and corrector scheme, Shampine and Watts [6] use the
formula as the predictor formula,
y p n+1 = 1 3 (y n−2 + y n−1 + y n ) + h 6 (3f n−2 − 4f n−1 + 13f n )
y p n+2 = 1 3 (y n−2 + y n−1 + y n ) + h 12 (29f n−2 − 72f n−1 + 79f n )
(6)
A P ECE scheme or P E(CE) k were implemented for a nonstiff problem.
3. Block Multistage and Multistep Method
Given a s stages and m steps integration formula [2], where h −1 ( ∑ m
j=0 kq j y i+s−j) approximates
y ′ (x i+q ) of order h m , q = 1, 2, ..., s. Consider a few formulas
1. for s=m=1, it is the BDF formula.
2. For s=2, m=3,
3. for s=3, m=5,
2y n+2 + 3y n+1 − 6y n + y n−1 = 6hf n+1
11y n+2 − 18y n+1 + 9y n − 2y n−1 = 6hf n+1
(7)
−3y n+3 + 30y n+2 + 20y n+1 − 60y n + 15y n−1 − 2y n−2 = 60hf(x n+1 , y n+1 )
12y n+3 + 65y n+2 − 120y n+1 + 60y n − 20y n−1 + 3y n−2 = 60hf(x n+2 , y n+2 )
137y n+3 − 300y n+2 + 300y n+1 − 200y n + 75 n−1 − 12y n−2 = 60hf(x n+3 , y n+3 )
(8)
These formulas can be implemented in either Newton iteration or byP ECE scheme, but a new predictor
need to be found first. For Equation (7), the predictor can still use Equation (6). Other predictors can
be found by using the similar idea as Adams-Bashforth formula, we will not give the exact formula here.
4. Stability regions
The main difficulty associated with stiff equations is that even though the component of the true
solutions corresponding to some eigenvalue that may becoming negligible, the restriction on the stepsize
imposed by the numerical stability of the method requires that |hλ| remain small throughout the range
of integration. So a suitable formula for stiff equations would be the one that would not require that |hλ|
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