Kulikov, G. Yu.; Kuznetsov, E. B.; Khrustaleva, E. Yu. On the global error control in nested implicit Runge-Kutta methods of Gauss type. (Russian, English) Zbl 1299.65172 Sib. Zh. Vychisl. Mat. 14, No. 3, 245-259 (2011); translation in Numer. Analysis Appl. 4, No. 3, 199-209 (2011). Summary: We develop the technique of an automatic global error control based on a combined step size and order control proposed by G. Yu. Kulikov and E. Yu. Khrustaleva [Comput. Math. Math. Phys. 48, No. 8, 1313–1326 (2008; Zbl 1199.65262)]. Special attention is given to the efficiency of computation because the implicit extrapolation based on the multi-stage implicit Runge-Kutta schemes might be expensive. Especially, we discuss the technique of global error estimation and control in order to compute the numerical solution satisfying the user-supplied accuracy conditions (in exact arithmetic) in the automatic mode. The theoretical results of this paper are confirmed by numerical experiments on test problems. Cited in 4 Documents MSC: 65L70 Error bounds for numerical methods for ordinary differential equations 65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems Keywords:implicit Runge-Kutta formula; effective implementation; nested implicit schemes of Gauss type; global error estimation; global error control; numerical experiments Citations:Zbl 1199.65262 PDFBibTeX XMLCite \textit{G. Yu. Kulikov} et al., Sib. Zh. Vychisl. Mat. 14, No. 3, 245--259 (2011; Zbl 1299.65172); translation in Numer. Analysis Appl. 4, No. 3, 199--209 (2011) Full Text: DOI