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| 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 the 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 s-stage doubly quasi-consistent parallel explicit peer methods of order s - 1 when s = 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. |
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