Variable-Stepsize Doubly Quasi-Consistent Parallel Explicit Peer Methods with Global Error Control

Applied Numerical Mathematics, 62(10) (2012), 1591-1603
http://dx.doi.org/10.1016/j.apnum.2012.06.018

Variable-stepsize explicit peer methods were designed and studied by Weiner et al. in 2008, 2009. Those schemes have proved their high efficiency in practical computations and are considered to be competitive to the best explicit Runge–Kutta embedded pairs. This paper adds more functionality to the mentioned numerical technique in terms of global error estimation and control. Theoretically, it is based on the new concept of double quasi-consistency introduced by Kulikov in 2009. This property means that the principal terms of the local and global errors coincide. In other words, the global error estimation and control can be done effectively via the conventional local error control facility, that is a standard feature of ODE solvers.

Recently, Kulikov and Weiner implemented the idea of double quasi-consistency in fixed-stepsize doubly quasi-consistent parallel explicit peer methods. They also extended that result to non-equidistant meshes by an accurate polynomial interpolation technique. Here, we prove at first that the class of variable-stepsize doubly quasi-consistent methods is not empty and provide the first sample of such numerical schemes. Then, we utilize the notion of embedded formulas to evaluate and control efficiently the local error of the constructed doubly quasi-consistent peer method and, hence, its global error at the same time. Numerical examples of this paper confirm that the usual local error control implemented in doubly quasi-consistent numerical integration techniques is capable of producing numerical solutions for user-supplied accuracy conditions in automatic mode. A comparison with the third order Matlab solver ode23 is also presented.