Math 6610 • Analysis of Numerical Methods I

{Wave Equation} Consider the wave equation in one space dimension:
\[
  u_{tt}=c^2u_{xx},\qquad c>0.
\]
Suppose \(0\le x\le1\) and \(t\ge0\). The problem requires initial and boundary
conditions but you can ignore them for the purposes of this question. Consider
the usual discretization
\[
  \frac{1}{k^2}\left(U_m^{n+1}-2U_m^n+U_m^{n-1}\right)
  =
  \frac{c^2}{h^2}\left(U_{m+1}^n-2U_m^n+U_{m-1}^n\right)
\]
where
\[
  U_m^n\approx u(x_m,t_n)
\]
and
\[
  x_m=mh,\qquad h=\frac{1}{M},\qquad
  \text{and}\qquad t_n=nk,\qquad h,k>0.
\]
Let the grid constant \(r\) be defined by
\[
  r=\frac{kc}{h}.
\]
Derive the constraint on \(r\) imposed by the Courant-Friedrichs-Lewy condition.
There is a value of \(r\) for which the local truncation error of the difference
scheme is zero. Derive that value.