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Gauss-type nested implicit Runge--Kutta methods exhibit many vital properties of implicit Runge--Kutta methods, such as stability, highorder accuracy, and symmetry. Moreover, nested implicit Runge--Kutta methods have a cheap practical implementation in comparison of implicit Runge--Kutta methods because of their explicit nature of internal stages. In this paper, we provide a complete mechanism for the construction, formulation, and implementation of 4th- and 6th-order nested implicit Runge--Kutta methods. The numerical testing is performed using 6th-order nested implicit Runge--Kutta method that uses an embedded 4th-order nested implicit Runge--Kutta method for the local error estimation on a collection of test problems from stiff detest. We also compare the nested implicit Runge--Kutta method with the MATLAB integrator ode 15s and assess the effectiveness of four variations on the local error estimation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the accuracy and efficiency of Gauss-type nested implicit Runge-Kutta (NIRK) methods. These methods possess many important practical properties of implicit Runge-Kutta (IRK) methods, such as, good stability, high order, and symmetry. Moreover, as with IRK methods, NIRK methods can also be presented in embedded form that helps local error estimation. However, the implementation of NIRK methods is practically cheaper than other classes of IRK methods as the NIRK formulas have explicit internal stages. We test an order six NIRK method that uses an embedded order four method for estimating the local error. Our test problems are those from stiff DETEST. As part of the testing, we compare the method with the Matlab integrator ODE15s and assess the effectiveness of four variations on the local error estimation. [ABSTRACT FROM AUTHOR]