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Unformatted text preview: sed relatively precise information to get the a priori
error bound of Example 2.1.3. We can expect less precision when confronted with
more realistic problems.
Error estimates that rely on the computed solution are called a posteriori estimates.
Can you design an a posteriori procedure for estimating either the local or global
errors of Euler's method? (Hint: you could try comparing solutions computed on
di erent meshes.)
2. Consider a linear ODE with variable coe cients
y 0 = () + ()
aty bt Consider IVPs with initial data 0 and 0 = 0 + 0 and show that the perturbed
solution n = n ; n,
0, of Euler's method satis es
y z y z y n = n n;1 + ( n 1)
ha t ; n;1 : Thus, we can identify = ( n 1) and apply the absolute stability condition (2.1.16)
locally. Similarly, show that the perturbed Euler solution of the nonlinear ODE
(2.1.1) satis es
at n = ; n;1 + ( h f tn;1 zn;1 ); ( n
ft ; 1 yn;1 )] : If is a smooth function of , show that
f y n = n;1 + h @f
t ; 1 yn;1 ) n;1 +(
O 2 n;1 )] : In this case, we can identify = y ( n 1 n 1) and, once again, determine the
region of absolute stability locally using (2.1.16). These heuristic arguments should
be established by rigorous means at some stage.
f t ; y ; 3. We've already observed that error estimates computed according to a priori bounds
such as (2.1.13) are too conservative to be used for practical step size control. Let
us consider an alternate method of estimating the global errors for Euler's method
that gives more precise information.
17 3.1. Assume that y exists and show that the local error of Euler's method satis es 000 dn 2 = h 2 y 00 3 ( n 1) +
t h 6 ; y 000 ( n) 2(n n t ; 1 ) tn : 3.2. Show that the global error
en = ( n) ; ( ( yt yn : satis es
en = en;1 + h en;1 fy tn;1 y tn;1 )) + 2 ( n 1) + ( 2)]
h y 00 t Oh ; ( 1], pp. 13-14).
3.3. Show that the above di erence equation is the Euler solution of the IVP
^ = ( ( ))^ + ( ) + ( )
2 de f dt tyt y e 00 t ^(0) = 0 Oh e where
^( ) = e tn en tn h = nh: Neglecting the ( ) term
Oh ^ = ( ( ))^ + ( )
dt f tyt e y 00 ^(0) = 0 t e : 3.4. ( 1], p. 24.) The solution of the above equation typically furnishes more precise
error information than (12). Use the solution of the above equation with the
a priori estimate (2.1.13) to calculate error estimates for the IVPs
y 0 =2 ty y 0 = ;2 0 ty <t< 1 y (0) = 1 4. ( 1], p. 24.) The solution of the IVP
0 x =; y x (0) = 1 y 0 = x y is the unit circle
( ) = cos xt t 18 ( ) = sin yt t: (0) = 0 : 4.1. Show that Euler's method
xn = xn;1 ; hyn;1 yn = yn;1 + hxn;1 does not form a closed curve, but, in fact, forms a spiral when = 2
h =N . 4.2. Show that the solution of
xn = xn;1 ; hyn;1 yn = yn;1 + hxn does form a closed curve and, hence, appears to provide a better approximation. (In each case, you may answer the question analytically or computationally.) 2.2 The implicit Euler method: Sti Systems
Consider the IVP
y 0 = ; ( ; 2) + 2
y t t t> 0 y (0) = y0 which has the solution
()= yt ; y0 e t + 2 t: Let us suppose that is a large positive real number. In...
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