Finite Difference and Shooting Methods for Two-Point Boundary Value Problems: A Comparative Analysis
Loading...
Date
2020-05-03
Journal Title
Journal ISSN
Volume Title
Publisher
MUST Journal of Research and Development (MJRD)
Abstract
Finite difference and shooting methods are popular with numerical practitioners in
solving two-point boundary value problems governed by ordinary differential equations.
However, the available literature is silent on which method is the most suitable with
respect to accuracy, efficiency, stability, and convergence. In this article, finite difference
and shooting methods are applied to solve numerically three types of two-point boundary
value problems. One problem is governed by a linear non-stiff differential equation, a
second problem is governed by a linear stiff differential equation, and a third problem is
governed by a non-linear differential equation. The analytical solution of each problem is
given. These solutions are used in assessing the accuracy attained by each of the
numerical methods. It is known a prior that finite difference schemes based on central
difference quotients in approximating derivative terms are numerically stable. Numerical
experiments carried out on the three problems lead to the conclusion that, for linear non-
stiff equations, linear shooting gives significantly more accurate results compared to
finite difference methods. However, in the case of linear stiff equations, the finite
difference method gives very accurate results while the shooting method fails totally,
displaying serious instability. As for problems governed by nonlinear equations, although
both finite difference and shooting methods converge, the shooting method converges
significantly faster than its counterpart. With respect to the attribute of efficiency, finite
difference methods are more efficient than shooting methods due to the ease with which
the Runge - Kutta initial value problem solver can be applied.
Description
This Journal Article was published by Mbeya University of Science and Technology in 2020