Math 6610 • Analysis of Numerical Methods I

{Wave Equation} For the PDE \(v_t+cv_x=0\), derive the Lax-Wendroff scheme
\[
  u_j^{n+1}=u_j^n
  -\frac{1}{2}\left(\frac{c\Delta t}{\Delta x}\right)
    (u_{j+1}^n-u_{j-1}^n)
  +\frac{1}{2}\left(\frac{c\Delta t}{\Delta x}\right)^2
    (u_{j-1}^n-2u_j^n+u_{j+1}^n).
\]
Analyze this scheme in terms of accuracy and stability. Does this scheme satisfy a
maximum principle? If it does, explain why. If it does not, give an example to show
this.