Locally Linearized Runge Kutta method of Dormand and Prince
For researchers solving initial value problems, this method offers improved efficiency and accuracy over standard Runge-Kutta methods, though it is an incremental improvement.
This paper introduces locally linearized Runge-Kutta formulas based on Dormand and Prince methods, demonstrating through numerical simulations on physical equations that they achieve significantly higher accuracy, reducing time steps and overall computation cost.
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 physical 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 computation cost of their adaptive implementation.