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\pagestyle{fancy} \lhead{\today}%02/04/2008}
\chead{POD-DEIM reduced system: Burgers'Equation}
\rhead{\bfseries Saifon} \lfoot{} \cfoot{}
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\section*{Finite Difference System of Burgers' Equation}
Consider again 1D unsteady Burgers'Equation:
\begin{eqnarray}
\frac{\partial}{\partial t} y(x,t) = \nu\frac{\partial^2}{\partial x^2}y(x,t)
- \frac{\partial}{\partial x}\left(\frac{y(x,t)^2}{2} \right)  &  &x \in [0,1], t \ge 0
\label{eqn:burgers}
\end{eqnarray}
\begin{eqnarray*}
y(0,t) = y(1,t) =0, t \ge 0  \  \textrm{ and } \
y(x,0) = y_0(x),   x \in [0,1],
\end{eqnarray*}
where $y(x,t)$ is the unknown function of time $t$
and location $x \in \Omega \equiv [0,1] $;
$\nu$ is a diffusion coefficient(viscosity parameter);
and $y_0(x)$ is an initial condition.
In this writeup, the initial condition is
$y_0(x) = f(x)-f(0)$, where $f(x) = e^{-(15(x-0.5))^2)}$;
$\nu= 0.1$; $t \in[0,1]$.
% and the number of finite difference (FD) is used for spatial discretization (on [0,1]) for the full-order system is 100. The POD basis functions are constructed from 70 snapshot solutions which are obtained by solving the full-order FD system at equally-spaced time steps in the interval $[0,2]$.
Finite difference (FD) for  spatial  discretization is given by
 \begin{eqnarray}
\frac{d}{d t} \by(t)=   \nu\bA \by(t) + \bF(\by) ,
\label{eqn:burgersFD}
\end{eqnarray}
where
$\bA \in \real^{n \times n}$ is the discrete Laplace operator;
$\bF(\by) = - \by.* \bA_x \by$ with first-order discrete differential operator $\bA_x \in \real^{n \times n}$
and `$.*$' denotes pointwise multiplication
(note: $-\frac{\partial}{\partial x}\left(\frac{y(x,t)^2}{2} \right) = -y(x,t) \frac{\partial y(x,t)}{\partial x}$).
Here the full-order dimension $n$ is 100.

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