Math 6610 • Analysis of Numerical Methods I

{Wave Equation}
Consider the Lax-Friedrichs scheme below for the PDE \(v_t+cv_x=0\), where you may
assume that \(c>0\):
\[
  u_j^{n+1}=\frac{1}{2}(u_{j-1}^n+u_{j+1}^n)
  -\frac{ck}{2h}(u_{j+1}^n-u_{j-1}^n),
\]
where \(h\) is the spacestep and \(k\) is the timestep. Note that the scheme can also be
written
\[
  u_j^{n+1}=u_j^n-\frac{ck}{2h}(u_{j+1}^n-u_{j-1}^n)
  +\frac{1}{2}(u_{j-1}^n-2u_j^n+u_{j+1}^n).
\]

\textbf{a)} Analyze the stability of this scheme in terms of \(\alpha=\frac{ck}{h}\).

\textbf{b)} Recall that a modified equation for a scheme is the PDE satisfied, to a
certain order, by a smooth solution \(u(x,t)\) of the difference scheme. To zeroth
order in \(k\) and \(h\), the modified equation for the Lax-Friedrichs scheme is
\(u_t+cu_x=0\). Derive the modified equation through terms of first order in \(k\) and
\(h\). You may assume that \(\alpha\) is constant. Does this modified equation predict
the same stability condition as your analysis in part \textbf{a)}? Explain your
answer. What does the modified equation suggest about the numerical solution that
will be obtained for the step-function initial data \(u(x,0)=1\) for \(x\le 0\) and
\(u(x,0)=0\) for \(x>0\)?