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# Some nal notes of comparison between runge kutta and

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Unformatted text preview: ethods unless the di erential system has rapidly oscillating solutions. The solution of the BEAM problem of Example 5.7.4 is oscillatory and many BDF codes failed on it. The Runge-Kutta code RADAU5 was the second-most e cient method on this problem. The A-stable method within the SPRINT package was, by far, the most e cient technique. Problems 1. Consider the solution of y0 = y + y2 t>0 y(0) = 1 which has the exact solution t y(t) = 1 + e; e t : This IVP is sti when Re( ) 0. In this case, the solution behaves like e t , i.e., like the solution of the linear problem y0 = y. Suppose that this problem is solved by the backward Euler method. 1.1. Find the maximum step size h for which functional iteration (5.7.2) converges when = ;104 . 1.2. Show that Newton's iteration converges for much larger step sizes. 64 Bibliography 1] U.M. Ascher and L.R. Petzold. Computer Methods for Ordinary Di erential Equations and Di erential-Algebraic Equations. SIAM, Philadelphia, 1998. 2] M. Berzins and R.M. Furzeland. A user's manual for sprint - a versatile software package for solving systems of algebraic, ordinary and partial di erential equations: Part 1 - algebraic and ordinary di erential equations. Technical report, Thornton Research Centre, Shell Research Ltd., Amsterdam, 1993. 3] W.E. Boyce and R.C. DiPrima. Elementary Di erential Equations and Boundary Value Problems. John Wiley and Sons, New York, third edition, 1977. 4] K.E. Brenan, S.L Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Di erential-Algebraic Equations. North Holland, New York, 1989. 5] P.N. Brown, G.D. Byrne, and A.C. Hindmarsh. Vode: a variable coe cient ode solver. SIAM J. Sci. Stat. Comput., 10:1039{1051, 1989. 6] R.L. Burden and J.D. Faires. Numerical Analysis. PWS-Kent, Boston, fth edition, 1993. 7] G.D. Byrne and A.C. Hindmarsh. A polyalgorithm for the numerical solution of ordinary di erential equations. ACM Trans. Math. Software, 1:71{96, 1975. 8] G. Dahlquist. A special stability problem for some linear multistep methods. BIT, 3:27{43, 1963. 9] J.R. Dormand and P.J. Prince. A f...
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