Locally Linearized Runge Kutta method of Dormand and Prince

TL;DR

This paper introduces embedded locally linearized Runge-Kutta formulas based on Dormand and Prince methods, demonstrating through numerical simulations that they significantly improve accuracy and reduce computational cost for initial value problems.

Contribution

The paper develops and analyzes a new locally linearized version of Dormand and Prince Runge-Kutta methods, showing enhanced performance over traditional formulas.

Findings

Locally linearized formulas achieve higher accuracy.

Significant reduction in number of time steps.

Lower overall computational cost.

Abstract

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.