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Presenting Error and ConvergenceIn the AMME2000 and AMME2960 units of study you will be asked to present the error andconvergence rates for different numerical methods to demonstrate the accuracy and performanceof the methods. This section will show how to calculate and present the error and convergence. Theexample problem used is calculation of the gradient of the functionf(x) =exusing the backwardand central difference schemes. This problem is the second task from the second week’s tutorial.To refresh, the backward difference scheme is:f0(x)≈f(x)-f(x-Δx)Δx(1)and the central difference scheme is:f0(x)≈f(x+ Δx)-f(x-Δx)2Δx(2)There are two ways that the error from these schemes could be calculated. The first is by performinga Taylor series expansion of the terms on the right-hand side and rearranging to isolate the errorterm. The second is by evaluating the two numerical schemes to find the approximate gradientof a function and then calculating the error from the difference between the true gradient of thefunction and the approximate gradient. The first method will give a mathematical expression forthe error when the scheme is applied to any function and the second will give a numerical valuefor the error when the scheme is applied to a particular function at a particular point or over aparticular domain. Both measures of error are valid and should be used where appropriate.You should use the Taylor series method if you are asked to:•Find or derive an expression for the error•Analytically obtain the order of accuracy of a scheme•Explain why or prove that a scheme possesses a certain order of accuracy or errorNote that the Taylor series is the appropriate method for finding an analytical expression for theerror from a finite difference scheme such as the backward or central schemes.Other numeri-cal methods may require different techniques to analytically demonstrate convergence and errorproperties.You should calculate the error by evaluating the numerical scheme for a particular problem if youare asked to:•Calculate the value of the error•Perform a convergence studyHere we will look at calculating the value of the error and performing a convergence study. Watchthe videos included with the material for the first week for the use of the Taylor series to find ananalytical expression for the error.Let’s begin by calculating the magnitudes of the errors from the backward and central differencesschemes at the pointx= 0.5.We’ll express the error as a percentage of the magnitude of theanalytical gradient:1
=|f0(x)-ef0(x)||f0(x)|×100(3)In this equationf0is the analytical gradient andef0is the numerical gradient. The subscriptsbandcwill be used for errors and gradients from the backward and central schemes respectively,e.g.