Unformatted text preview: ough RungeKutta formulas are tedious to derive, we can make a few general
observations. An order one formula must be exact when the solution of the ODE is a
linear polynomial. Were this not true, it wouldn't annihilate the constant and linear
10 terms in a Taylor's series expansion of the exact ODE solution and, hence, could not
have the requisite O(h2) local error to be rstorder accurate. Thus, the RungeKutta
method should produce exact solutions of the di erential equations y = 0 and y = 1.
The constantsolution condition is satis ed identically by construction of the RungeKutta formulas. Using (3.2.3a), the latter (linearsolution) condition with y(t) = t and
f (t y) = 1 implies
s
X
tn = tn 1 + h bi
0 ; i=1 or
s
X
i=1 0 bi = 1: (3.2.5a) If we also require the intermediate solutions Yi to be rst order, then the use of (3.2.3b)
with Yi = tn 1 + cih gives
s
X
ci = aij
i = 1 2 : : : s:
(3.2.5b)
; j =1 This condition does not have to be satis ed for loworder RungeKutta methods 16]
however, its satisfaction simpli es the task of obtaining order conditions for higher...
View
Full
Document
This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.
 Spring '14
 JosephE.Flaherty

Click to edit the document details