Programming Assignment Three

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}} \item{1a.} Consider Burgers equation $$u_t+2u u_x=0\quad \hbox{where $x\in[0,2\pi]$ and $t\ge 0$}$$ with initial condition $u(x,0)=u_0(x)$ and periodic boundary conditions $u(0,t)=u(2\pi,t)$. Non-zero solutions to this equation develop shock discontinuities in finite time at which point they become non-unique. A physically relevant solution may be obtained as the limit of solutions to the viscous Burgers equation $$u_t-\nu u_{xx}+2u u_x=0$$ as $\nu\to 0$. Solve the viscous Burgers equation with $u_0(x)=2+3\sin x$ for $\nu=0.5$, $0.25$ and $0.125$. Compare the graphs of $u(x,1)$ for each of these values of $\nu$. Do these graphs converge? Can you compute a solution for $\nu=0.01$? What about for $\nu=0.001$?

\item{1b.} Consider the semi-discrete upwind method $$ {d u_j\over dt}=-{2 u_j\over{\Delta x}} \cases{ u_j - u_{j-1} & for $u_j\ge 0$\cr u_{j+1}-u_j & for $u_j<0$.\cr} $$ Write a program using Euler's explicit method to solve these equations. Compare the solutions obtained by this method to the solutions found in part a. Draw graphs of $u(x,1/8)$, $u(x,1/4)$, $u(x,1/2)$ and $u(x,1)$. What is the speed of propagation of the shock? Does the speed of propagation obey the Rankine--Hugoniot condition?

\item{1c.} Write Burgers equation in the form of a hyperbolic conservation law $$ u_t + \partial_x f(u) =0 $$ and show for a smooth solution that $$ \int_0^{2\pi} u(x,t) dx =\int_0^{2\pi} u(x,0) dx. $$ The discrete analog of this is $$ \sum_{j=1}^{n} u^{m}_j = \sum_{j=1}^n u^0_j. $$ A numerical method for which the above property holds for all $m$ is called conservative. Is the numerical scheme in part b conservative?

\item{1d.} Define the averages $$ \tilde f^m_{j+1/2}={1\over \Delta t}\int_{t_m}^{t_{m+1}} f\big(u(x_{j+1/2},t)\big)d t$$ and $$ \tilde u^m_j={1\over \Delta x} \int_{x_{j-1/2}}^{x_{j+1/2}} u(x,t_m) dx. $$ Derive the exact mathematical identity $$ \tilde u_j^{m+1}=\tilde u_j^{m}-{\Delta t\over \Delta x} \big(\tilde f^m_{j+1/2}-\tilde f^m_{j-1/2}\big) $$ by integrating the hyperbolic conservation law $u_t+\partial_x f(u)=0$ over the intervals $[t_m,t_{m+1}]$ and $[x_{j-1/2},x_{j+1/2}]$ in time and space.

\item{1e.} The Lax--Wendroff method is obtained by making the approximation $$ \tilde f^m_{j+1/2}=f\Big( {u^m_j+u^m_{j+1}\over 2} +{\Delta t\over 2\Delta x} \big\{ f\big(u^m_j)-f(u^m_{j+1})\big\}\Big). $$ Write a program to solve Burgers equation using the Lax--Wendroff method. Compare the graphs of $u(x,1/8)$, $u(x,1/4)$, $u(x,1/2)$ and $u(x,1)$ to those found in part a. Do these solutions obey the Rankine--Hugoniot condition? Is this method conservative?

\item{1f.} The Engquist--Osher method is obtained by making the approximation $$%\eqalign{ \tilde f^m_{j+1/2}= \cases{ f(u^m_{j+1})& $f'(u^m_{j})\le 0$, $f'(u^m_{j+1})<0$\cr f(\xi)& $f'(u^m_{j})\le 0$, $f'(u^m_{j+1})\ge 0$\cr f(u^m_{j})+f(u^m_{j+1})-f(\xi)& $f'(u^m_{j})>0$, $f'(u^m_{j+1})<0$\cr f(u^m_{j})& $f'(u^m_{j})>0$, $f'(u^m_{j+1})\ge 0$\cr } $$ where $\xi$ in the second and third cases is the number between $u^m_{j}$ and $u^m_{j+1}$ such that $f'(\xi)=0$. Write a program to solve Burgers equation using the Engquist--Osher method. Compare the graphs of $u(x,1/8)$, $u(x,1/4)$, $u(x,1/2)$ and $u(x,1)$ to those found in part e. Which method resolves the shocks better? Are there spurious oscillations?

Last Updated: Sun Apr 27 02:31:28 PDT 2003