Math 6610 • Analysis of Numerical Methods I

{Wave Equations}
Consider the first-order wave equation in one space dimension:
\[
  u_t+au_x=0,\qquad -\infty<x<\infty,\qquad t>0,
\]
where \(a\) is a constant, with the initial condition
\[
  u(x,0)=u_0(x),\qquad -\infty<x<\infty
\]
for a given function \(u_0\). We use the usual discretization
\[
  U_j^n\approx u(x_j,t^n),
\]
where \(x_j=j\Delta x\) and \(t^n=n\Delta t\). For a finite difference scheme to solve
this problem, what is the CFL condition? Give an intuitive explanation for the
condition. Derive the first-order upwind scheme for this equation and its truncation
error. For numerical stability, derive the conditions for stability in the maximum
norm (maximum principle) and the \(\ell_2\) norm (Fourier analysis).