Programming Assignment Two

Your work should be presented in the form of a typed report using clear and properly punctuated English. Where appropriate include full program listings and output. If you choose to work in a group of two, please turn in independently prepared reports.

\def\R{{\bf R}} \def\Z{{\bf Z}} \hangindent\parindent \item{1a.} Consider the viscous Burgers equation $$u_t-u_{xx}+2u u_x=0\quad \hbox{where $x\in[0,2\pi]$ and $t\ge 0$}$$ with initial condition $u(x,0)=u_0(x)$ and periodic boundary conditions $u(0,t)=u(2\pi,t)$. Derive the semi-discrete spectral method $$ {d \hat u_k\over dt} + k^2 \hat u_k + \sum_{m=-\min(n,n-k)}^{\min(n,n+k)} ik \hat u_{k-m} \hat u_m = 0 $$ by making the the approximation $$u(x,t)\approx\sum_{k=-n}^n \hat u_k(t) e^{ikx}.$$

\item{1b.} Find a standard subroutine for performing a fast Fourier transform. Use your mouse to connect to the National Institute of Standards and Technology web page at{\tt\ http://math.nist.gov/}. From this page select the following links in order \smallskip {\narrower \item{1.}Guide to Available Mathematical Software \item{2.}What Problem it Solves \item{3.}Taxonomy as a Decision Tree \item{4.}J Integral Transforms \item{5.}J1 Trigonometric Transforms \item{6.}J1a One-dimensional \item{7.}J1a2 Complex \smallskip}\hangindent\parindent How many routines are there to perform a fast Fourier transform? Download one of them and print it out.

\item{1c.} Let ${\cal F}$ denote the Fourier transform on the interval $[0,2\pi]$. The convolution theorem may be used to obtain that $$ \eqalign{ \sum_{m=-\min(n,n-k)}^{\min(n,n+k)} ik \hat u_{k-m} \hat u_m &=ik {\cal F}\big\{({\cal F}^{-1} \hat u )^2\big\}_k. }$$ Compare the number operations needed to evaluate the sum directly with the number of operations needed to evaluate it by means of a fast Fourier transform. Note that the Fourier transform method obtains the sum for all values of $k$ simultaneously. How does the total computational cost of each method depend on $n$?

\item{1d.} Write a program to solve $$ {d \hat u_k\over dt} + k^2 \hat u_k + ik {\cal F}\big\{({\cal F}^{-1} \hat u )^2\big\}_k = 0 $$ using Euler's explicit method. Let $\hat u_k(0)={\cal F}\{8e^{(\pi-x)^2}\}_k$ and draw accurate graphs of $u(x,1/8)$, $u(x,1/4)$, $u(x,1/2)$ and $u(x,1)$.

\item{1e.} Consider the semi-discrete central difference method $$ {d u_j\over dt} -{1\over (\Delta x)^2}(u_{j-1}-2u_j+u_{j+1}) +{1\over \Delta x}u_j(u_{j+1}-u_{j-1})=0. $$ By taking $u_{n}=u_{0}$ and $u_{-1}=u_{n-1}$ this method may also be used to compute periodic solutions. Write a program using Euler's explicit method to solve these equations. Let $u(x,0)=8e^{(\pi-x)^2}$ on $[0,2\pi]$ and compare the graphs of $u(x,1/8)$, $u(x,1/4)$, $u(x,1/2)$ and $u(x,1)$ with those found in the previous step.

\item{1f.} [Extra Credit] Replace Euler's explicit method for the time integration by the explicit Runga-Kutta method of order four given by $$ \eqalign{ K_0&=\Delta t\, F(u^m)\cr K_1&=\Delta t\, F(u^m+K_0/2)\cr K_2&=\Delta t\, F(u^m+K_1/2)\cr K_3&=\Delta t\, F(u^m+K_2)\cr u^{m+1}&=u^m+{1\over 6}(K_0+2K_1+2K_2+K_3)} $$ and find $u(3\pi/2,1/8)$ accurately to 3 decimal places.

Last Updated: Thu Apr 10 04:21:09 PDT 2003