Solving Lorenz System by Using Lower Order Symmetrized Runge-Kutta Methods

Abstract

Runge-Kutta is a widely used numerical method for solving the non-linear Lorenz system. This study focuses on solving the Lorenz equations with the classical parameter values by using the lower order symmetrized Runge-Kutta methods, Implicit Midpoint Rule (IMR), and Implicit Trapezoidal Rule (ITR). We show the construction of the symmetrical method and present the numerical experiments based on the two methods without symmetrization, with one- and two-step active symmetrization in a constant step size setting. For our numerical experiments, we use MATLAB software to solve and plot the graphical solutions of the Lorenz system. We compare the oscillatory behaviour of the solutions and it appears that IMR and two-step active IMR turn out to be chaotic while the rest turn out to be non-chaotic. We also compare the accuracy and efficiency of the methods and the result shows that IMR performs better than the symmetrizers, while two-step active ITR performs better than ITR and one-step active ITR. Based on the results, we conclude that different implicit numerical methods with different steps of active symmetrization can significantly impact the solutions of the non-linear Lorenz system. Since most study on solving the Lorenz system is based on explicit time schemes, we hope this study can motivate other researchers to analyze the Lorenz equations further by using Runge-Kutta methods based on implicit time schemes.