Math 6610 • Analysis of Numerical Methods I

{Local Truncation Error} Consider the wave equation
\[
  u_{tt}=c^2u_{xx}
  \tag{7}
\]
and its discretization
\[
  x_m=mh,\qquad t_n=nk,\qquad U_m^n\approx u(x_m,t_n)
  \tag{8}
\]
and
\[
  U_m^{n+1}-2U_m^n+U_m^{n-1}
  =
  r\left(U_{m+1}^n-2U_m^n+U_{m-1}^n\right)
  \tag{9}
\]
where \(r=c^2k^2/h^2\) is the grid constant. For the purposes of this problem,
ignore the issues of initial and boundary conditions. Compute the local
truncation error of (9) and show that there is a value of \(r\) for which the
local truncation error is exactly zero. Give a physical interpretation of this
fact.