Abstract
In considering explicit Runge-Kutta methods to find starting values for linear multi-step methods in solving numerically first-order differential equations, the minimization of truncation error is of utmost importance. The criterion chosen to minimize the truncation error is minimizing the sum of the magnitudes of the coefficients of the principal error function. The parameters in the Runge-Kutta formulas permit this. Further, for fourth- and fifth-order Runge-Kutta methods one must choose between working with a single differential equation or a system because the principal error functions are different. In this dissertation the choice is to work with systems of differential equations. For both third- and fourth-order methods the optimum formula is found. The third-order formula is not new, but the fourth-order formula is. But for fifth-order Runge-Kutta, the optimum formula is found for only the classes of formulas resulting from Lobatto, Newton-Cotes, Radau, and Legendre-Gauss quadrature numbers. The optimum formula is of Lobatto type. The restriction on the classes of formulas considered is made because otherwise the number of parameters seems to make the error function unmanageable.
Crawley, Alton Rudolph (1970). Optimum runge-kutta formulas of third-, fourth-, and fifth-orders. Doctoral dissertation, Texas A&M University. Texas A&M University. Libraries. Available electronically from
https : / /hdl .handle .net /1969 .1 /DISSERTATIONS -177147.