The Runge-Kutta class of iterative methods is designed to approximate solutions of a system of ordinary differential equations (ODE). The second-order class of Runge-Kutta methods is determined by a system of 3 nonlinear equations and 4 unknowns, and includes the modified-Euler and mid-point methods. The fourth-order class is determined by a system of 8 nonlinear equations and 10 unknowns. This work formulates the question of identifying good values of these 8 parameters for a given family of ODE as a blackbox optimization problem. The objective is to determine the parameter values that minimize the overall error produced by a Runge-Kutta method on a training set of ODE. Numerical experiments are conducted using the NOMAD direct-search optimization solver.
Published March 2017 , 11 pages