Math 6610 • Analysis of Numerical Methods I

{Wave Equations}
Consider the standard wave equation in one space dimension:
\[
  u_{tt}=c^2u_{xx},\qquad 0<x<1,\quad t>0,
\]
with initial conditions
\[
  u(x,0)=f(x),\qquad u_t(x,0)=g(x),\qquad 0<x<1
\]
for given smooth functions \(f\) and \(g\), and boundary condition
\[
  u(0,t)=u(1,t)=0,\qquad t>0.
\]
The leap-frog scheme takes advantage of the fact that the equation can be
reformulated as a system of first order equations
\[
  u_t+cv_x=0,
  \qquad
  v_t+cu_x=0.
\]
Denote \(x_j=j\Delta x\), \(U_j^n\approx u(x_j,n\Delta t)\), and
\(V_{j+1/2}^{n+1/2}\approx v(x_{j+1/2},(n+\frac{1}{2})\Delta t)\); use central
difference approximations for both derivatives to derive the leap-frog scheme for
the system of equations. How do you implement the initial and boundary conditions
for both \(u\) and \(v\)? Derive the truncation error for this approximation and
determine the order of truncation error. For stability, consider solutions in the
form
\[
  \begin{bmatrix}
    U^n\\
    V^{n+1/2}
  \end{bmatrix}
  =
  \lambda^n e^{ikx}
  \begin{bmatrix}
    \widetilde{U}\\
    \widetilde{V}
  \end{bmatrix},
\]
for some constants \(\widetilde{U}\) and \(\widetilde{V}\). Substitute this solution
into the finite difference equations, find out what condition \(\lambda\) has to
satisfy in order for \(\widetilde{U}\) and \(\widetilde{V}\) to have nontrivial
solutions. For those \(\lambda\) values, compute \(\lvert\lambda\rvert\) to see if the
stability condition \(\lvert\lambda\rvert\le 1\) is satisfied.