The Computable Differential Equation Lecture ... - Bruce E. Shapiro
The Computable Differential Equation Lecture ... - Bruce E. Shapiro The Computable Differential Equation Lecture ... - Bruce E. Shapiro
124 CHAPTER 6. LINEAR MULTISTEP METHODS 6.4 Adams Methods Definition 6.3. An Adams Method is a linear multistep method with a 0 = 1, a 1 = −1, and a k = 0 for all k > 1. The explicit Adams-Bashforth Methods are given by y n = y n−1 + h(b 1 f n−1 + b 2 f n−2 + · · · + b k f n−k ) (6.72) while the implicit Adams-Moulton Methods y n = y n−1 + h(b 0 f n + b 2 f n−1 + · · · + b k f n−k ) (6.73) Adams methods are derived by integrating the differential equation over two grid points ∫ tn y n = y n−1 + f(s, y(s))ds (6.74) t n−1 and approximating the integrand with an interpolating polynomial over the past several mesh points. The interpolating polynomial of choice is the Newton backward difference formula (BDF), where P k (t n ) = f(t n ) + ( s = k) k∑ ( ) (−1) j −s ∇ j f(t j n ) (6.75) j=1 s(s − 1) · · · (s − k + 1) k! (6.76) t = t n + sh (6.77) ∇p n = p n − pn − 1 (6.78) ) ∇ k p n = ∇ (∇ k−1 p n , k ≥ 2 (6.79) The resulting Adams method coefficients are then given by ∑k−1 ( ) ∫ b j = (−1) j−1 i 1 ( ) (−1) i −s ds (6.80) j − 1 i i=j−1 Example 6.2. Derive the 3-stage Adams-Bashforth method. Solution. The method will be given by y n = y n−1 + ∫ tn 0 t n−1 f(u, y(u))du (6.81) where f is estimated using ( ) ( ) −s −s f ≈ f n−1 + (−1) ∇f 1 n−1 + (−1) 2 ∇ 2 2 f n−1 ( ) (6.82) −s +(−1) 3 ∇ 3 3 f n−1 Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 6. LINEAR MULTISTEP METHODS 125 Since ( ) (−1) 1 −s = (−1)(−s) = s (6.83) 1 ( ) (−1) 2 −s 2 (−s)(−s − 1) = (−1) = 1 2 2 2 (s2 + s) (6.84) ( ) (−1) 3 −s 3 (−s)(−s − 1)(−s − 2) = (−1) = 1 3 6 6 (s3 + 3x 2 + 2s) (6.85) Newton’s backward difference formula becomes f ≈ f n−1 + s∇f n−1 + 1 2 (s2 + s)∇ 2 f n−1 + 1 6 (s3 + 3s 2 + 2s)∇ 3 f n−1 (6.86) Changing the variable of integration from t to s = (t − t n−1 )/h, equation 6.81 becomes y n = y n−1 + Substituting the approximation formula, y n = y n−1 + h ∫ 1 0 ∫ 1 0 f(s, y(s))ds (6.87) [ f n−1 + s∇f n−1 + s2 + s ∇ 2 f n−1 + (6.88) 2 s 3 + 3s 2 ] + 2s ∇ 3 f n−1 ds (6.89) 6 = y n−1 + h [f n−1 + 1 2 ∇f n−1 + 5 12 ∇2 f n−1 + 3 ] 8 ∇3 f n−1 (6.90) [ = y n−1 + h f n−1 + 1 2 (f n−1 − f n−2 ) + 5 12 (f n−1 − 2f n−2 + f n−3 )+ (6.91) ] 3 8 (f n−1 − 3f n−2 + 3f n−3 − f n−4 ) (6.92) [ 55 = y n−1 + h 24 f n−1 − 59 24 f n−2 + 37 24 f n−3 − 3 ] 8 f n−4 (6.93) Example 6.3. Derive the 3-step Adams-Moulton implicit formula. Solution. The method is similar to the Adams-Bashforth derivation except for the following two modifications: the interpolation formula is evaluated at f n instead of f n−1 ; and because the integration interval precedes the point where the interpolation c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
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CHAPTER 6. LINEAR MULTISTEP METHODS 125<br />
Since<br />
( )<br />
(−1) 1 −s<br />
= (−1)(−s) = s (6.83)<br />
1<br />
( )<br />
(−1) 2 −s<br />
2 (−s)(−s − 1)<br />
= (−1) = 1 2<br />
2 2 (s2 + s) (6.84)<br />
( )<br />
(−1) 3 −s<br />
3 (−s)(−s − 1)(−s − 2)<br />
= (−1) = 1 3<br />
6<br />
6 (s3 + 3x 2 + 2s) (6.85)<br />
Newton’s backward difference formula becomes<br />
f ≈ f n−1 + s∇f n−1 + 1 2 (s2 + s)∇ 2 f n−1 + 1 6 (s3 + 3s 2 + 2s)∇ 3 f n−1 (6.86)<br />
Changing the variable of integration from t to s = (t − t n−1 )/h, equation 6.81<br />
becomes<br />
y n = y n−1 +<br />
Substituting the approximation formula,<br />
y n = y n−1 + h<br />
∫ 1<br />
0<br />
∫ 1<br />
0<br />
f(s, y(s))ds (6.87)<br />
[<br />
f n−1 + s∇f n−1 + s2 + s<br />
∇ 2 f n−1 + (6.88)<br />
2<br />
s 3 + 3s 2 ]<br />
+ 2s<br />
∇ 3 f n−1 ds (6.89)<br />
6<br />
= y n−1 + h<br />
[f n−1 + 1 2 ∇f n−1 + 5 12 ∇2 f n−1 + 3 ]<br />
8 ∇3 f n−1 (6.90)<br />
[<br />
= y n−1 + h f n−1 + 1 2 (f n−1 − f n−2 ) + 5 12 (f n−1 − 2f n−2 + f n−3 )+ (6.91)<br />
]<br />
3<br />
8 (f n−1 − 3f n−2 + 3f n−3 − f n−4 )<br />
(6.92)<br />
[ 55<br />
= y n−1 + h<br />
24 f n−1 − 59<br />
24 f n−2 + 37<br />
24 f n−3 − 3 ]<br />
8 f n−4<br />
(6.93)<br />
Example 6.3. Derive the 3-step Adams-Moulton implicit formula.<br />
Solution. <strong>The</strong> method is similar to the Adams-Bashforth derivation except for the<br />
following two modifications: the interpolation formula is evaluated at f n instead of<br />
f n−1 ; and because the integration interval precedes the point where the interpolation<br />
c○2007, B.E.<strong>Shapiro</strong><br />
Last revised: May 23, 2007<br />
Math 582B, Spring 2007<br />
California State University Northridge
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