Construction of explicit runge-kutta pairs with stiffness detection

K e y w o r d s O r d i n a r y differential equations, Initial value problems, Runge-Kut ta methods, Stiffness detection, Symbolic computation, Computer algebra systems, Computer generation of numerical methods. 1. I N T R O D U C T I O N A framework for explicit Runge-Kutta methods is being implemented as part of an ongoing overhaul of MATHEMATICA~S differential equation solver NDSolve. One of our goals was to establish a unified environment for a whole class of methods of different orders. This helps to provide a uniform basis for comparing methods, reduces the potential for programming errors, and allows the optimization of a single implementation from which all methods benefit. Furthermore, specification of additional Runge-Kutta methods can be accomplished by simply entering the appropriate coefficients. It is well known that explicit Runge-Kutta methods are not suitable for the numerical solution of stiff differential equations. One of the features that we wanted to incorporate in our implementation was automatic stiffness detection. In this way users are provided with run-time information about when the choice of method is inappropriate. It also becomes possible to switch between stiff and nonstiff Runge-Kutta methods [1]. 0895-7177/05/$ see front mat ter (~) 2005 Elsevier Ltd. All rights reserved. Typeset by AA/tS-~X doi:10.1016/j.mcm.2005.01.01