Adaptive nested implicit Runge-Kutta formulas of Gauss type

Applied Numerical Mathematics, 59 (2009), 707-722
http://dx.doi.org/10.1016/j.apnum.2008.03.019

This paper deals with a special family of implicit Runge–Kutta formulas of orders 2, 4 and 6. These methods are of Gauss type; i.e., they are based on Gauss quadrature formulas of orders 2, 4 and 6, respectively. However, the methods under discussion have only explicit internal stages that lead to cheap practical implementation. Some of the stage values calculated in a step of the numerical integration are of sufficiently high accuracy that allows for dense output of the same order as the Runge–Kutta formula used. On the other hand, the methods developed are A-stable, stiffly accurate and symmetric. Moreover, they are conjugate to a symplectic method up to order 6 at least. All of these make the new methods attractive for solving nonstiff and stiff ordinary differential equations, including Hamiltonian and reversible problems. For adaptivity, different strategies of error estimation are discussed and examined numerically.