Development and Analysis of Advanced Numerical Algorithms for Solving Differential Equations Using Taylor, Euler, and Runge-Kutta Methods

Abstract

This paper describes the creation, implementation and theoretical analysis of three fundamental numerical methods related to ordinary differential equations (ODE), these are the third and higher order Taylor Series method, the Euler Method (note that we are only considering the implicit version), and the Runge-Kutta family of algorithms, specifically the classical fourth-order Runge-Kutta method (RK4). We specifically consider the fundamental trade-off of accuracy vs cost for numerical solvers. We will create an organized implementation of these methods, we will perform an in-depth theoretical study of these methods by considering properties such as local truncation error, global error, convergence and stability properties using the linear test equation. We will also perform a full empirical study employing these algorithms on the set of classical test problems that include linear, non-linear and stiff problems. In the final conclusions we hope to demonstrate that while the high-order Taylor methods can yield us excellent accuracy for super-smooth functions, the requirement to compute analytical derivatives is a huge disadvantage. While the Euler method is simple, it is neither efficient nor stable. The Runge-Kutta methods - most prominently RK4 - are more superior in that they offer an overall higher efficiency, stability, and simplicity than other counterpart methods. And the main contribution of this work is to provide a clear, well-justified insights about when each method can be expected to be useful, its strengths and weaknesses, and I hope to contribute a useful awareness for both practitioners and educators.