Unformatted text preview: with RungeKutta methods indicates that implicit methods have better stability properties than
explicit methods. Thus, with the objectives of developing implicit methods having good
stability properties for sti systems and of illustrating another class of methods, we'll
23 examine backwarddi erence formulas. These methods are derived by approximating y(t)
by a k thdegree interpolating polynomial Pk (t) using the k +1 points tn;i, i = 0 1 : : : k,
and di erentiating Pk0 (t) to approximate y0(t). Once again, we'll use the Newton form of
the interpolating polynomial which may be obtained from (5.3.3) by replacing n ; 1 by
n, f by y, and k ; 1 by k to obtain
k
i;1
X riy(tn) Y
Pk (t) =
(t ; tn;j )
i
i=0 i!h j =0
k
y(k+1)( ) Y(t ; t )
Ek (t) = y(t) ; Pk (t) = (k + 1)!
n;j
j =0 (5.5.1a) 2 (tn;k tn): (5.5.1b) The backwarddi erence method will be obtained by collocating at t = tn , i.e., by
enforcing Pk0 (tn) = f (tn Pk (tn)): (5.5.2) Once again, it will be convenient to change variables by letting
= t ; tn
h
and use (5.3.4) to obtain
i;1
1 Y(t ; t ) = ( + 1) : : : ( + i ; 1) = (;1)i ;
n;j
i
i!hi j=0
i! (5.5.3a) : Then, (5.5.1) can be written as Pk (t) = k
X (;1)i ;
i
i=0 riy(tn) (5.5.3b) and Ek (t) = (;1)k+1hk+1 k; 1 y(k+1)( ):
+ (5.5.3c) Di erentiating (5.5.3b) P 0 (t
k k
1 dPk (0) = 1 X riy(t )
n
n) =
hd
h i=1 i 24 (5.5.4a) where
i = (;1)i dd ; i =0 ( + 1)( + 2) : : : ( + i ; 1)
i! = dd i 1: =0 Di erentiating the product yields
=1
i i i 1: (5.5.4b) Di erentiation of the error expression (5.5.3b) proceeds in the same manner. Bear in
mind that is a function of t (or ) however, ; k+1 =0 =0 so
k
( (tn)) = h y(k+1)( (tn)):
k+1
Replacing y(tn) in (5.3.1) by Pk (tn) + ek (tn) and using (5.5.4) yields 0
Ek (tn) = hk k+1 y (k+1) (5.5.4c) k
X ri y(tn) hk+1 (k+1)
i + k + 1 y ( ) = hf (tn y(tn)):
i=1 Neglecting the local discretization error yields the backwarddi erence formula
k
X riyn
i = hf (tn yn):
i=1 (5.5.5a) Its local discretization error is
n= k
k
X riy(tn)
; hf (tn y(tn)) = ; k h 1 y(k+1)( )
i
+...
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 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

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