A Comparative Analysis of Forward, Backward and Central Finite Difference Schemes for Numerical Differentiation of Transcendental Function
Abstract
Yusupha Bah, Bai Mbye Cham and Dawda Charreh
Research has shown that not all mathematical equations can be solved analytically. Numerical methods provide effective approaches for obtaining approximate solutions to complex mathematical problems that cannot be solved analytically using traditional pen-and-paper techniques. In this research, we investigate one of the most commonly used numerical methods, namely, the finite difference method. This is a numerical technique used to approximate the values of derivatives using discrete data points. The finite difference method consists of three main schemes: the forward difference, backward difference, and central difference methods. In this study, we compare these three finite difference schemes to determine which one converges fastest over a range of step sizes. Transcendental functions, including the exponential function (ex ) and the trigonometric function sin(x), are used to evaluate and compare the performance and convergence of the three schemes. MATLAB is used to analyse the relationship between the step size h and the truncation and round-off errors by examining how changes in the step size affect the convergence rate of each scheme. The results show that the central difference scheme is significantly more accurate than the forward and backward difference schemes, demonstrating second-order convergence. However, when the step size becomes excessively small, round-off error, sometimes associated with the turning-point phenomenon, becomes significant. Consequently, instead of continuing to decrease, the numerical error begins to increase. Based on these findings, researchers seeking higher accuracy and faster convergence in numerical differentiation should consider using the central difference scheme, particularly when an appropriate step size is selected.
