辅导案例-MECH 5315M-Assignment 2
MECH 5315M { Engineering Computational Methods Assignment 2 Copyright R 2019 University of Leeds UK. All rights reserved. Please submit electronically via Minerva before noon Tuesday 28 January 2020. Your report should not be longer than 4 pages in total. Please use at least a font size of 11pt and 2cm margins on all sides. All MATLAB code should be attached to the report as an appendix. The appendix does not count towards the 4 page limit. All gures in the report must have captions, properly labelled axes, legends where necessary and must be described and analysed in the text. Problem. In this exercise, we will study a combination of the transport and heat equation called the advection-diusion equation _u(x; t) + v 0 u x (x; t) = u xx (x; t): (1) For all numerical examples, please consider 0 x 2, a nal time of t end = 2:0 and a value of v 0 = 1:5. Task 1. (25 marks) 1. (3 5 = 15 marks) Derive the initial value problem in spectral space that arises when solving (1) using the pseudo-spectral method. Proceed along the following steps: a) How does the equation for the residual R(u(x; t)) look like that comes out of plugging the truncated Fourier series u(x; t) = 1 N N 1 X k=0 u^ k (t)e ikx into (1)? b) What are the N equations that result from enforcing R(u(x n ; t)) = 0 at N equidistant mesh points x n = 2n N ; n = 0; : : : ; N 1? (2) 1 c) How can those N equations be written compactly using the spectral dierentiation matrix D? You do not need to comment on the \trick" we used the rewrite the second part of D and may simply ignore the issue. However, in your MATLAB code, be sure to use the matrix D that is dened in the provided examples. 2. (5 marks MATLAB plus 5 marks explanation) Write a MATLAB script that solves the initial value problem coming out of Task 1.1 for = 0:1 using ode45 for u 0 (x) = cos(3x) and then plots the solution at the end of the simulation in both physical and spectral space using N = 64. How does the solution change for an initial value u 0 (x) = cos(7x)? Describe how the solution behaves. You do not need to explain the results here, this will be done in Task 2.5. Task 2. (30 marks) 1. (5 marks) Apply the continuous Fourier transform in x to Equation (1). Write down the resulting equation in spectral space. Make sure to clearly explain what identities from Fourier analysis you have used to obtain your result. 2. (5 marks) Write down the general solution of the dierential equation arising from Task 1.1. 3. (5 marks) Find a text book that tells you what the Fourier transform of cos(3x) is. Write down the result and cite your source. 4. (5 marks) How does the solution from Task 1.2 look like for initial values u(x; 0) = cos(3x) and u(x; 0) = cos(7x)? 5. (10 marks) Based on your results from 2.4, explain your observations in Task 1.2. Task 3. (30 marks) 1. (5 marks) Write a MATLAB function that solves Equation (1) using the pseudo-spectral method and explicit Euler and returns the approximate solution at the end of the simu- lation in physical space. The function should have input variables N (number of Fourier modes), t end (nal time), v 0 (transport velocity), (viscosity) and u 0 (initial value in physical space) . 2. (5 marks) Write a MATLAB function that solves Equation (1) using centred nite dif- ferences for both u x and u xx and explicit Euler and returns the approximate solution at the end of the simulation. The function should have input variables N x (number of nite dierence points), t end (nal time), v 0 (transport velocity), (viscosity) and u 0 (initial value in physical space). You can copy the code to generate the needed nite dierence matrices from the examples provided in Minerva. 3. (5 marks) Write a MATLAB script that calls your functions from 3.1 and 3.2 for initial values u (1) 0 (x) = exp