Explicit Runge-Kutta methods

Explicit Runge-Kutta methods are some of the most popular time stepping methods applied by scientists and engineers. They are simple to program and are self starting.13; Just the initial conditions suffice. These methods are available to different orders of accuracy, the classical fourth order method being the most popular among these.13; In spite of their popularity, most books and references do not appear to document them in full. Derivations are given mostly for the second order methods. Stability is often skipped. In this document, the explicit methods up to fourth order are derived and all known solutions are documented in such a way that it is easy to pick a desired method. A13; new set of coefficients for the fourth order method is also included. For the sake of completeness, the basics of stability are discussed and the linear stability diagrams13; are reproduced. A brief discussion from recent developments on nonlinear stability of explicit Runge-Kutta methods is also included. Implementation of the methods using C++ and an object oriented design is described. C++ templates form the basis for this design