Approximate solution of ordinary differential equations and their systems through discrete and continuous embedded Runge-Kutta formulae and upgrading of their order (Q1339288)

The author presents a summary of his research on Runge-Kutta (RK) algorithms for the approximate solution of ordinary differential equations. At first he derives a method for the computer generation of the algebraic equations for the coefficients of RK algorithms. Algorithms are found in which other RK algorithms are embedded. Several families of 5th order 6-stage algorithms of this type are obtained, one of which is given explicitly here. The presence of an embedded algorithm is used for a priori error assessment and interpolation purposes. The author develops three 6th order 8-stage formulas and a family of 8th order 13-stage formulas which are applicable to both scalar and coupled systems of differential equations. Recently the author has turned his attention to modified RK algorithms whose output are \(C^ 1\) curves embedded in the ordinary differential equation trajectories. A set of algorithms is presented in each of which a \(C^ 1\) curve, which approximates the solution trajectory to 4th order, is generated from a 5th order 6-stage algorithm. These algorithms have promised to be useful in future applications. Application of RK algorithms to ordinary differential equations of higher order is also discussed briefly.