Consider the linear advection equation
where
![$x \in [0,10]$](img2.gif)
with periodic boundary conditions.
Consider also two different initial conditions
Compute the solution on a 128-point mesh for both initial conditions and for CFL = 0.8. Compare the first- and second-order upwind schemes versus the Lax-Wendroff scheme after 100 time steps and after 10,000 time steps.
![\begin{eqnarray*}\mbox{{\rm (a)}}\;\;\;\; \Theta_0 (x,0) & = & \sin (2\pi x)\\
...
...1, x\in [1,2]\\
0, \mbox{{\rm elsewhere}}\end{array}\right .
\end{eqnarray*}](img3.gif)