errorConvergenceGuide.pdf - Presenting Error and Convergence In the AMME2000 and AMME2960 units of study you will be asked to present the error and

errorConvergenceGuide.pdf - Presenting Error and...

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Presenting Error and Convergence In the AMME2000 and AMME2960 units of study you will be asked to present the error and convergence rates for different numerical methods to demonstrate the accuracy and performance of the methods. This section will show how to calculate and present the error and convergence. The example problem used is calculation of the gradient of the function f ( x ) = e x using the backward and central difference schemes. This problem is the second task from the second week’s tutorial. To refresh, the backward difference scheme is: f 0 ( x ) f ( x ) - f ( x - Δ x ) Δ x (1) and the central difference scheme is: f 0 ( x ) f ( x + Δ x ) - f ( x - Δ x ) x (2) There are two ways that the error from these schemes could be calculated. The first is by performing a Taylor series expansion of the terms on the right-hand side and rearranging to isolate the error term. The second is by evaluating the two numerical schemes to find the approximate gradient of a function and then calculating the error from the difference between the true gradient of the function and the approximate gradient. The first method will give a mathematical expression for the error when the scheme is applied to any function and the second will give a numerical value for the error when the scheme is applied to a particular function at a particular point or over a particular 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 error Note that the Taylor series is the appropriate method for finding an analytical expression for the error 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 error properties. You should calculate the error by evaluating the numerical scheme for a particular problem if you are asked to: Calculate the value of the error Perform a convergence study Here we will look at calculating the value of the error and performing a convergence study. Watch the videos included with the material for the first week for the use of the Taylor series to find an analytical expression for the error. Let’s begin by calculating the magnitudes of the errors from the backward and central differences schemes at the point x = 0 . 5. We’ll express the error as a percentage of the magnitude of the analytical gradient: 1
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= | f 0 ( x ) - e f 0 ( x ) | | f 0 ( x ) | × 100 (3) In this equation f 0 is the analytical gradient and e f 0 is the numerical gradient. The subscripts b and c will be used for errors and gradients from the backward and central schemes respectively, e.g.
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