Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation

Recently, Kulikov presented the idea of double quasi-consistency, which facilitates global error estimation and control, considerably. More precisely, a local error control implemented in such methods plays a part of global error control at the same time. However, Kulikov studied only Nordsieck formulas and proved that there exists no doubly quasi-consistent scheme among those methods.Here, we prove that the class of doubly quasi-consistent formulas is not empty and present the first example of such sort. This scheme belongs to the family of superconvergent explicit two-step peer methods constructed by Weiner, Schmitt, Podhaisky and Jebens. We present a sample of ss-stage doubly quasi-consistent parallel explicit peer methods of order s−1s−1 when s=3s=3. The notion of embedded formulas is utilized to evaluate 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 clearly 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.