Comparing Runge-Kutta methods with Lotka-Volterra model : accuracy, computational efficiency, and stiffness in ODEs

Abstract

This thesis compares the second-order and fourth-order Runge-Kutta methods in their application to the Lotka-Volterra model, through ordinary differential equations. The study entails a comprehensive theoretical examination of the Runge-Kutta methods, backed by numerical experiments to evaluate their performance. The fundamental goal is to identify which method better balances computational efficiency with accuracy in the presence of stiffness between second and fourth-order Runge-Kutta methods. The study evaluates the methods' computational cost and accuracy with large step sizes to determine their suitability for long-term simulations and systems with differing degrees of stiffness.

The results demonstrate that fourth-order Runge-Kutta method, despite its higher computational load due to four function evaluations per step, provides a significant advantage in maintaining accuracy over larger step sizes. This makes it preferable for extensive simulations where precision is critical, especially in stiff scenarios. Conversely, second-order Runge-Kutta method, with only two function evaluations, is less computationally demanding and is more suited to systems with lower stiffness, where the trade-off between speed and accuracy is less severe. The thesis concludes that while both second and fourth-order Runge-Kutta methods are effective for solving ordinary differential equations, fourth-order Runge-Kutta method's higher-order accuracy makes it the superior choice for modeling the dynamics of the Lotka-Volterra model when dealing with stiff systems.