The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 5. RUNGE-KUTTA METHODS 103
The complete derivation of the order conditions makes use of graph-theoritic
methods that are beyond the scope of this presentation (see [7]). Instead, we quote
some of the simpler results. For example, a method must satisfy the following if it
is of order p:
b T C k−1 1 = 1 , k = 1, 2, . . . , p (5.126)
k
b T A k−1 1 = 1 , k = 1, 2, . . . , p (5.127)
k!
where C = diagonal(c 1 , . . . , c s ), b T = (b 1 , . . . , b s ), A is the matrix of coefficients a ij ,
and s is the number of stages. Since ∑ k a jk = c j we have the additional condition
A1 = c, where c = (c 1 , . . . , c s ) T . Expanding to individual components we have the
following order conditions. For order 1,
b T 1 = ∑ i
b i = 1 (5.128)
For order 2, in addition to the order 1 conditions, we must have
b T c = ∑ i
b i c i = 1 2
(5.129)
For order 3 we must also have the following conditions:
b T Cc = ∑ i
b i c 2 i = 1 3
(5.130)
b T Ac = ∑ i,j
b i a ij c j = 1 6
(5.131)
The upper limit of these sums in 5.128 through 5.131 is the number of stages, so there
are different conditions depending on the number of stages in the methods. The formulas
have been automated in the Mathematica package NumericalMath‘Butcher‘
as the function RungeKuttaOrderConditions. To access this function enter