An analysis similar to the one used in theorem 211

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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 @y (n 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 ^ = ( ( ))^ + ( ) + ( ) y 2 de f dt tyt y e 00 t ^(0) = 0 Oh e where ^( ) = e tn en tn h = nh: Neglecting the ( ) term Oh ^ = ( ( ))^ + ( ) y 2 de 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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