Doubly quasi-consistent fixed-stepsize numerical integration of stiff ordinary differential equations with implicit two-step peer methods

Journal of Computational and Applied Mathematics, 340 (2018), 256-275
https://doi.org/10.1016/j.cam.2018.02.037

The problem of accurate and efficient numerical integration of ordinary differential equations (ODEs) is a key issue of computation science, which has a valued practical impact in various topics of research. In this respect, effective global error estimation and control strategies are of particular theoretical and practical significance. They allow many applied dynamic systems to be treated numerically for user-supplied accuracy conditions in automatic mode. The property of double quasi-consistency plays an outstanding role in the mentioned topic because the global (or true) error of a doubly quasi-consistent method is asymptotically equal to its local error. Therefore, this entails quite a nontrivial implication that the true error of numerical integration can be cheaply evaluated and regulated within a single solution run. The latter was considered to be impossible for decades. The property of double quasi-consistency goes back to 2009, when Kulikov studied it in the first time but failed to find doubly quasi-consistent numerical methods among Nordsieck formulas. Later on, Kulikov and Weiner proved the existence of doubly quasi-consistent numerical schemes among superconvergent explicit two-step parallel peer methods and examined their practical efficiency in comparison to explicit Matlab ODE solvers in 2010. The focus of the present research is on accurate numerical integration formulas for treating stiff ODEs, which often arise in practice and for which explicit methods are shown to be ineffective. Here, we make the first step towards an accurate and efficient numerical solution of stiff ODEs and prove existence of implicit doubly quasi-consistent formulas. We fulfill our investigation of double quasi-consistency within the family of fixed-stepsize implicit two-step peer schemes and construct two methods of convergence orders 3 and 4, which possess excellent stability properties. Our theoretical results are confirmed with numerical integration of both nonstiff and stiff test problems in this paper.