In this paper we present new singly diagonally implicit two-step peer triples equipped with local and global error controls for providing preassigned accuracies of numerical integration of stiff ordinary differential equations (ODEs) in automatic mode. Recently, Kulikov and Weiner [A singly diagonally implicit two-step peer triple with global error control for stiff ordinary differential equations, SIAM J. Sci. Comput., 2015, 37(3), A1593-A1613] reported an efficient numerical integration tool of order 2, which solves accurately many difficult stiff ODEs, including large-scale systems obtained from semidiscretization of partial differential equations (PDEs), corresponding to user-supplied requests. Moreover, the cited method is not only accurate, but it is also more efficient than the well-known Matlab code ODE23s with local error control. Here, we further extend the published technique and construct variable-stepsize singly diagonally implicit two-step peer triples of the higher orders 3 and 4. Our numerical experiments suggest that these triples are suitable for treating stiff problems with prescribed accuracy conditions. In addition, performance of the presented methods can be comparable to the built-in stiff Matlab code ODE15s with local error control, which is considered to be a benchmark means for solving stiff ODEs by many practitioners, for some test problems.Keywords: Ordinary differential equation, stiff problem, singly diagonally implicit two-step peer method, absolute and scaled local and global error estimations, automatic local and global error controls 2000 MSC: 65L05, 65L06, 65L20, 65L50, 65L70.

Summary.For the numerical solution of initial value problems of ordinary differential equations partitioned adaptive Runge-Kutta methods are studied. These methods consist of an adaptive Runge-Kutta method for the treatment of a stiff system and a corresponding explicit Runge-Kutta method for a nonstiff system. First we modify the theory of Butcher series for partitioned adaptive Runge-Kutta methods. We show that for any explicit Runge-Kutta method there exists a translation invariant partitioned adaptive Runge-Kutta method of the same order. Secondly we derive a special translation invariant partitioned adaptive Runge-Kutta method of order 3. An automatic stiffness detection and a stepsize control basing on Richardson-extrapolation are performed. Extensive tests and comparisons with the partitioned R K F 4 R Walgorithm from Rentrop [-16] and the partitioned algorithm LSODA from Hindmarsh [9] and Petzold [15] show that the partitioned adaptive RungeKutta algorithm works reliable and gives good numerical results. Furthermore these tests show that the automatic stiffness detection in this algorithm is effective.

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