Locally Linearized Runge Kutta method of Dormand and Prince

In this paper, the effect that produces the Local Linearization of the embedded Runge-Kutta formulas of Dormand and Prince for initial value problems is studied. For this, embedded Locally Linearized Runge-Kutta formulas are defined and their performance is analyzed by means of exhaustive numerical simulations. For a variety of well-known test equations with different dynamics, the simulation results show that the locally linearized formulas exhibit significant higher accuracy than the original ones, which implies a substantial reduction of the number of time steps and, consequently, a sensitive reduction of the overall computational cost of their adaptive implementation.