Assignment 1 AMME2000.docx - Elton Cheng SID 460390873...

Elton Cheng SID: 460390873
Assignment 1 AMME2000
Abstract
The report demonstrates the operations of different analysis techniques. In the report, both
analytical solutions and numerical solutions are tested for accuracy and validity in their
applications. Section 1 compares the results obtained from differentiation and various finite
difference stencils. Section 2 models the diffusion of gas in chamber using the Fourier’s
series.
Table of Contents
Introduction …………………………………………………………………1
Section 1 …………………………………………………………………1
Section 2 …………………………………………………………………4
Conclusion …………………………………………………………………5
Appendix …………………………………………………………………6
Introduction
Engineers attempt to implement theories from science to arrive at a solution that defines the
physical world and apply it to make life more convenient. During an engineering analysis,
obtaining an exact solution is not always s plausible, the solution may be difficult or
impossible to derive due to irregular and unpredictable nature of the physical world.
Fortunately, engineers came up with estimations to approximate the exact solutions. Section1
of the report examines the accuracy of these approximation through a simple comparison
between the values solved by second order differentiation and two different finite different
stencils. Section 2 uses a Fourier’s Series to derive an equation that imitates the diffusion of
gas in a chamber.
Section 1
The second differentiation of the function f(x) = cosh(2x) was analysised by using the
analytical method and numerical method between the interval of 0 ≤ x ≤ 1. The exact solution
for the second differentiation is f(x) = 4cosh(2x) and the numerical solution is calculated
using the two stencils:
Figure 1.1: Two finite difference stencils
The interval is divided into 4 points and the second differentiation solution of each point is
determined through both analytical and numerical method and is plotted into the graph as
shown below.
The deviance of the 1 st order plot and 2 nd order
plot from the exact plot in Figure 2
demonstrates that estimated value is from 1
st
order is less accurate from 2
nd
order. 2nd order
plot appears to have overlapped the exact plot
which signifies its accuracy. Further analysis is
made through the derivation of leading order
error of the two stencils.