The RK-Nystrøm Method
In “A History Of Runge Kutta Methods” (Applied Numerical Mathematics, 20, 1996, pp 247-260), J.C. Butcher presents a set of coefficients for a 5th order RK method as derived by Kutta. The tableau is shown below:
However, Butcher indicates that Kutta’s 5th order coefficients had slight errors, which were subsequently corrected by Nystrøm in 1925. The method with these coefficients are more commonly known as the RungeKutta – Nystrøm method. The tableau for the Nystrom coefficients are :
The Generalized RK formulation can then be used to solve a set of ODE’s.
Generalized Runge Kutta Formulation
For RK methods of order 4 and higher, the inter-stage vectors (k’s) can be represented using the following formulation (Atkinson et al, Numerical Solution of ODE’s):
The solution at t(n+1) is given by
J.C. Butcher (Numerical Methods for ODE’s, 2nd Edition) invented the “Butcher tableau” methodology of representing the coefficients a(i,j), b(j) and c(j) for the Runge Kutta methods. The general format of such a tableau is
The Butcher tableau for the most popular RK4 method thus becomes:
The use of the Butcher tableau, along with the generalized formulation from Atkinson, provides an easy methodology for numerical integration of ODEs.
A Handful Of Numerical Integration Techniques 
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