Construction and implementation of highly stable two-step continuous methods for stiff differential systems

We describe a class of two-step continuous methods for the numerical integration of initial-value problems based on stiff ordinary differential equations (ODEs). These methods generalize the class of two-step Runge-Kutta methods. We restrict our attention to methods of order p = m, where m is the number of internal stages, and stage order q = p to avoid order reduction phenomenon for stiff equations, and determine some of the parameters to reduce the contribution of high order terms in the local discretization error. Moreover, we enforce the methods to be A-stable and L-stable. The results of some fixed and variable stepsize numerical experiments which indicate the effectiveness of two-step continuous methods and reliability of local error estimation will also be presented.