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# 14 that we failed to solve by eulers method because

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Unformatted text preview: series expansion to show that a similar ratio of successive solutions of the backward Euler method satis es Re h yt yt h ; e e : ; e yn = h 0 1 ; (h2 h (1 =h) which is a very good approximation of the exact ratio of solutions at successive time steps when 0. yn;1 e )2 < <h ; e h In a similar manner, the solution ratio for the explicit Euler method (2.1.2) is yn yn;1 =1+ which is clearly a poor approximation of 26 e h h when h 0. 2. The local error estimate (2.2.6) can be used to establish global convergence of the backward Euler method. In particular, follow the reasoning of Theorem 2.1.1 to establish following result. Theorem 2.2.1. Suppose ( ) is continuously di erentiable on fty f( ) j 0 ty t ;1 T <y< 1g : Then the backward Euler method (2.2.3) converges to the solution of (2.2.1) at a rate of O(h). 3. Consider the trapezoidal rule yn = yn;1 +2 (n h ft yn )+ ( n ft yn;1 1 ; )] (2.2.7) for the solution of the IVP (2.2.1). 3.1. Find expressions for its local discretization error and the local error. 3.2. Determine its region of absolute stability. 3.3. In order to simplify the iterative solution of (2.2.7), consider the \predictorcorrector" version of the trapezoidal rule ^= yn yn = yn;1 yn;1 + ( hf tn;1 yn;1 + 2 ( n ^n) + ( n h ft y ft ) 1 ; (2.2.8a) yn;1 )] : (2.2.8b) Thus, the forward Euler method (2.2.8a) is used to \predict" a solution ^n at n and the trapezoidal rule (2.2.7b) is used to correct it. The corrector (2.2.8b) could be iteratively to \correct" the solution again, but let's suppose that this is not done. y t 3.4. Determine the region of absolute stability of the predictor-corrector pair (2.2.8a,b) and compare it with the regions of absolute stability of the forward Euler method (2.1.2) and the trapezoidal rule (2.2.8). (It may be convenient to eliminate the predicted solution ^n from (2.2.8b) using (2.2.8a).) y 27 28 Bibliography 1] C.W. Gear. Numerical Initial Value Problems in Ordinary Di erential Equations. Prentice Hall, Englewood Cli s, 1971. 2] E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Di erential Equations I: Nonsti Problems. Springer-Verlag, Berlin, second edition, 1993. 29...
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