Automatic error control in the Gauss-type nested implicit Runge-Kutta formula of order 6

Russian Journal of Numerical Analysis and Mathematical Modelling, 24(2) (2009), 123-144
http://dx.doi.org/10.1515/RJNAMM.2009.009

Five different error estimation strategies suitable for the Gauss-type Nested Implicit Runge–Kutta method of order 4 have been presented and tested numerically in [Kulikov and Shindin, Lecture Notes in Computer Science: 136–143, 2007, Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. The nested Implicit Runge–Kutta schemes introduced recently are an efficient class of Implicit Runge–Kutta formulas. In this paper we deal with the methods of order 6. One scheme of such sort has been constructed in [Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. Now we present a one-parametric family of the above-mentioned formulas of order 6 by relaxing the accuracy requirement for some stage values. This allows the error estimation strategies designed for the method of order 4 to be extended to the higher-order Gauss-type Nested Implicit Runge–Kutta method. We also present the particulars of the efficient implementation of this method, which is stable and accurate. The numerical examples confirm the efficiency of the numerical scheme under consideration for both ordinary differential equations and partial differential equations.