An introduction to ``Almost Runge-Kutta'' methods (Q1360554)

The author introduces a new class of 4-stage methods of order \(p=4\) which are close to corresponding explicit Runge-Kutta methods but multivalue in character. Their basic idea is similar to that of using off-step points, e.g.\ one point from the previous step. The new methods are set up in such a way that the stage order is greater than achievable with Runge-Kutta methods, that the stability function is as simple as for Runge-Kutta methods and that very little information is passed between steps. The latter feature is important for economic stepsize changes while the first one is favourable for dense output extensions. In the paper the coefficients of four methods are given. An error control algorithm and an interpolatory dense output scheme is developed. Numerical experiments with the DETEST set of test problems are reported and a comparison with the classical Runge-Kutta method is made.