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 RungeKutta code RADAU5 was the secondmost e cient method
on this problem. The Astable 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 erentialAlgebraic 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 InitialValue
Problems in Di erentialAlgebraic 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. PWSKent, 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
 Spring '14
 JosephE.Flaherty
 Numerical Analysis, The Land, yn, Tn, Numerical ordinary differential equations

Click to edit the document details