Math 6610 • Analysis of Numerical Methods I

{Wave Equations}
Consider the wave equation in one space dimension:
\[
  u_t+au_x=0,\qquad -\infty<x<\infty,\qquad t>0,
\]
with initial condition
\[
  u(x,0)=u_0(x).
\]
Consider the usual discretization
\[
  U_m^k\approx u(x_m,t_k)
\]
where \(x_m=m\Delta x\) and \(t_k=k\Delta t\). For an explicit scheme, what is the CFL
condition? Give an intuitive explanation of the condition. Give the first-order
upwind scheme for this equation and derive the truncation error.