Numerical tests with Gauss-type nested implicit Runge-Kutta formulas

In this paper we conduct a detailed numerical analysis of the Gauss-type Nested Implicit Runge-Kutta formulas of order 4, introduced by Kulikov and Shindin in [4]. These methods possess many important practical properties such as high order, good stability, symmetry and so on. They are also conjugate to a symplectic method of order 6 at least. All of these make them efficient for solving many nonstiff and stiff ordinary differential equations (including Hamiltonian and reversible systems). On the other hand, Nested Implicit Runge-Kutta formulas have only explicit internal stages, in the sense that they are easily reduced to a single equation of the same dimension as the source problem. This means that such Runge-Kutta schemes admit a cheap implementation in practice. Here, we check the above-mentioned properties numerically. Different strategies of error estimation are also examined with the purpose of finding an effective one.