One-leg variable-coefficient formulas for ordinary differential equations and the local-global step size control

Numerical Algorithms , 43(1) (2006), 99-121
http://dx.doi.org/10.1007/s11075-006-9043-5

In this paper we discuss a class of numerical algorithms termed one-leg methods. This concept was introduced by Dahlquist in 1975 with the purpose of studying nonlinear stability properties of multistep methods for ordinary differential equations. Later, it was found out that these methods are themselves suitable for numerical integration because of good stability. Here, we investigate one-leg formulas on nonuniform grids. We prove that there exist zero-stable one-leg variable-coefficient methods at least up to order 11 and give examples of two-step methods of orders 2 and 3. In this paper we also develop local and global error estimation techniques for one-leg methods and implement them with the local–global step size selection suggested by Kulikov and Shindin in 1999. The goal of this error control is to obtain automatically numerical solutions for any reasonable accuracy set by the user. We show that the error control is more complicated in one-leg methods, especially when applied to stiff problems. Thus, we adapt our local–global step size selection strategy to one-leg methods.