Efficient Nested Implicit Runge–Kutta Methods for Nonlinear Stiff Problems

Abstract

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 4 th- and 6 th-order nested implicit Runge–Kutta methods. The numerical testing is performed using 6 th - order nested implicit Runge–Kutta method that uses an embedded 4 th - 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 ode15s and assess the effectiveness of four variations on the local error estimation.