Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Numerical investigations on global error estimation for ordinary differential equations
ICCAM '96 Proceedings of the seventh international congress on Computational and applied mathematics
Numerical methods for ordinary differential equations in the 20th century
Journal of Computational and Applied Mathematics - Special issue on numerical anaylsis 2000 Vol. VI: Ordinary differential equations and integral equations
On Global Error Estimation and Control for Initial Value Problems
SIAM Journal on Scientific Computing
Explicit two-step peer methods
Computers & Mathematics with Applications
Superconvergent explicit two-step peer methods
Journal of Computational and Applied Mathematics
On quasi-consistent integration by Nordsieck methods
Journal of Computational and Applied Mathematics
Doubly quasi-consistent parallel explicit peer methods with built-in global error estimation
Journal of Computational and Applied Mathematics
Variable-Stepsize Interpolating Explicit Parallel Peer Methods with Inherent Global Error Control
SIAM Journal on Scientific Computing
Global Error Control in Adaptive Nordsieck Methods
SIAM Journal on Scientific Computing
Adaptive ODE solvers in extended Kalman filtering algorithms
Journal of Computational and Applied Mathematics
Local and global error estimation and control within explicit two-step peer triples
Journal of Computational and Applied Mathematics
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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.