FOOD FOR THOUGHT: A FEW NUMERICAL DELICACIES

ABSTRACT:

Many learners when they take an elementary differential equations course for the first time, bring with them misconceptions from numerical [i]modus operandi[/i]s that they had learnt in their calculus courses, greatest in quantity notable of which concerns the interstice width in using a numerical [i]modus operandi[/i] It is important that we strive to dispel any of these misconceptions as well as to make scholars aware that there is ofttimes a trade-off between the complexity of a numerical manner and its computational cost. In this note, we provide examples to do that.

KEYWORDS: Differential equations, first-order initial value vexed question numerical methods, the Euler process the Runge-Kutta method, errors, computational costs

1 INTRODUCTION

Two manners that many introductory textbooks [1 2 3] discuss are the Euler rule and the 4th-order Runge-Kutta course Some introductory textbooks [1] also discuss a scarcely any other methods, such as the backward Euler [i]modus operandi[/i] the Adams-Bashforth method, the Adams-Moulton manner and, more generally, predictor-corrector manners and adaptive methods. In this note, we restrict our discussion to the Euler mode and the 4th-order Runge-Kutta method

When solving (1) numerically, individual has to be careful not to be misled through the numerical solution, for errors may arise in cunning ways. For example, it may be that the accumulated round-off error enlarges large or that the solution is unfasteneded in the interval of interest. In general, not many problems arise when [function of] is linear, on the contrary things become trickier when [function of] is nonlinear. In the latter case, a careful mix of analytical, graphical, and numerical work may provide useful information about what is happening in a particular solution. diocese [1, 4] for a discussion upon errors.

An important aspect of any numerical means is the mesh width that is used, for it influences the couple the error and the computational require to be paid [i]or[/i] undergone Due to the limited mark of a typical elementary differential equations textbook its discussion upon the errors and the computational require to be paid [i]or[/i] undergones associated with a numerical way and the importance of the choice of a interstice width may be very basic; upon the other hand, the presentation of these topics in a typical numerical analysis textbook may be rather technical, involving the use of Taylor series, for instance. In this note, we search for to bridge these two horizontals of discussion on errors and computational take away froms by providing simple, concrete examples that illustrate clearly in what manner a chosen mesh width may affect the two This note is organized as follows

In Section 3 we reinforce Fox's [5] point that what may be considered to be a small interstice width in one instance may not be small in another. We present to view in Section 4 that plane when the graph of a numerical solution captures the overall behavior of the exact solution, the numerical solution may provide a poor approximation of the solution itself. We also illustrate in that section a trade-off between the complexity of a numerical rule and its computational cost: a rule that is less complex to implement may incur more computational require to be paid [i]or[/i] undergone than a method that is more compounded to implement. Finally, we illustrate in Sections 5 and 6 that, in fact, in more [i]or[/i] less cases no mesh width may be small enough to allow a numerical [i]modus operandi[/i] to capture the behavior of the exact solution upon the solution's entire domain.

Today, scholars can easily explore IVPs of the stamp we discuss here graphically using a computer algebra combination of parts to form a whole like Matlab or even a programmable graphing calculator like the Texas Instruments TI-89 [8] The figures that appear here were generated using Matlab 6; the add-on package dfield6.m [7] was used to generate Figure 5(a). Like Fox [5] we tender that our students program the numerical meanss themselves; the Matlab code we used is available upon the Web at http://math.ucdavis.edu/~1hong/num_sol_ode.

2 THE NUMERICAL METHODS

A derivation can be set in many elementary differential equations textbooks; single derivation using Simpson's rule can be fix in [3], for example. The 4th-order Runge-Kutta course may be thought of as a weighted average of the function [function of] evaluated at different points.

It should not be a surprise that the 4th-order Runge-Kutta means belongs to what is called the Runge-Kutta class of methods; however, it may be a surprise that the Euler [i]modus operandi[/i] also belongs to the Runge-Kutta class of way s and is, in fact, the 1st-order Runge-Kutta means in the class. The 4th-order Runge Kutta means was the method that was originally discloseed by Runge and Kutta, and it is oftentimes simply referred to as the Runge-Kutta manner [1].

We remark that the order of the way refers to its truncation error [4 p 331](or global truncation error [1])1: the Euler rule being a 1st-order method, has a truncation error that is a constant times the pace size, h, and the Runge-Kutta [i]modus operandi[/i] being a 4th-order method, has a truncation error that is a constant times h^sup 4^ Loosely speaking, the error in an nth-order numerical rule reduces by 1/2^sup n^ each time the mesh width is halved. Hence, loosely speaking, the error in the Euler means is reduced by 1/2 each time the mesh width is halved, and the error in the Runge-Kutta manner is reduced by 1/2^sup 4^ = 1/16 each time the mesh width is halved. diocese [1, 4], for example, for a discussion upon errors.