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Unformatted text preview: s said to converge to order p. The computations are performed on a sequence of meshes having ner-and- ner spacing, but always ending at the same time . Although the de nition has been stated with
sequences of uniform meshes, it could easily be revised to include more general mesh
With these preliminaries behind us, we are in a position to state and prove our main
result. Once again, a Lipschitz condition on ( ) will come to our aid.
T fty 7 Theorem 2.1.1. Suppose ( ) exists and ( ) satis es a Lipschitz condition on the
strip f( ) j 0
1g. Then Euler's method (2.1.2) converges to the
y ty t 00 t T fty <y< solution of the IVP (2.1.1). Proof. From (2.1.4b), the exact solution of the IVP (2.1.1) satis es ( ) = ( n 1) + y tn yt ; ( ( hf tn;1 y tn;1 )) + dn : Subtracting this from (2.1.2) and using (2.1.3)
= en en;1 + ( ( h f tn;1 y tn;1 )) ; ( n
ft yn;1 1 ; )] + (2.1.7) dn : Since ( ) satis es a Lipschitz condition,
fty j (n
ft ; 1 ( y tn;1 )) ; ( n
ft yn;1 1 ; )j j( Ln y tn;1 ); yn;1 j = nj n 1j
L e ; : Taking the absolute value of (2.1.7) and using the triangular inequality and the above
Lipschitz condition yields j nj (1 +
e hLn )j en;1 Let
nN L and write (2.1.8) as Ln d j + j nj
d = 1max j nj
d j nj (1 + )j n 1j +
e Since the inequality holds for all hL (2.1.8) : e ; d: n j nj (1 + ) (1 + )j n 2 j + ] + = (1 + )2 j n 2j + 1 + (1 + )]
e hL hL e d ; d hL e ; d hL Continuing the iteration j nj (1 + )n j 0j + 1 + (1 + ) +
e hL e d hL ::: + (1 + Using the formula for the sum of a geometric series X(1 +
k=0 )k = (1 + hL 8 hL )n ; 1 hL hL )n 1 ]
; : : yields j nj (1 + )nj 0j +
e hL (1 + d e hL )n ; 1] hL : (2.1.9) Equation (2.1.9) can be written in a more revealing form by using the following
Lemma, which is common throughout numerical analysis. Lemma 2.1.1. For all real z 1+ (2.1.10) z z e: Proof. Using a Taylor's series expansion
2 1+ + 2
z z = e 2 (0 ) z e z: Neglecting the positive third term on the left leads to (2.1.10).
Now, using (2.1.10) with =
z , hL (1 +
since nh = tn T for n hL )n e hLn LT e . Using the above expression in (2.1.9) N j nj LT e e j 0j + d e hL ( LT e ; 1) (2.1.11) : Since ( ) exists for 2 0 ], Euler's method is consistent and, using (2.1.4c), we
can bound on as
y 00 t t T d 2 d = 0max j nj = 2 0max j ( )j
h d y 00 t (2.1.12) : Substituting (2.1.12) into (2.1.11) j nj LT e e Roundo error is neglected, so e0 j 0 j + 2 ( LT ; 1) 0max j ( )j
h e L = (0) ;
y e y0 y h L e y Thus, limh 0 j nj ! 0 and Euler's method converges.
! e 9 t : = 0, and we have j nj 2 ( LT ; 1) 0max j ( )j
e 00 00 t : (2.1.13) Remark 1. The assumption that y (t) exists is not necessary. A proof without this
assumption appears in Hairer et al. 2], Section 1.7. They also show that the Lipschitz
condition on f (t y) need only be satis ed on a compact domain rather than an in nite
strip. Gear 1], Section 1.3, presents a proof with the additional assumption that f (t y)
satisfy a Lipschitz condition on t instead of requiring y (t) to be bounded.
Example 2.1.3. Consider the simple problem...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
- Spring '14