{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Some nal notes of comparison between runge kutta and

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}