# How to obtain an implicit finite difference scheme for the wave equation?

Suppose I had the following problem:

$U_{tt}=U_{xx}+U_{yy}$ in $\Omega=[0,1]\times[0,1]$
$U(x,y,0)=f(x,y)$
$U_{t}(x,y,0)=g(x,y)$
$U=0$ on $\partial \Omega$

I know that there is an explicit finite difference scheme to solve this problem of the form:

$\frac{U(i,j,k+1) - 2U(i,j,k)+U(i,j,k-1)}{\Delta t^2} = \frac{U(i+1,j,k) - 2U(i,j,k)+U(i-1,j,k)}{\Delta x^2} + \frac{U(i,j+1,k) - 2U(i,j,k)+U(i,j-1,)}{\Delta y^2}$

by using a centered finite difference in time. My mind is thinking that it's possible to simply change the k's to k+1's to obtain an implicit scheme, but I haven't worked out the ensuing truncation error or stability analysis. My guess is that it's likely to be wrong... How do I derive an implicit scheme for this PDE?

• Why do you want to use an implicit scheme for this problem? Please remember what the FAQ says: You should only ask practical, answerable questions based on actual problems that you face. – David Ketcheson Jan 16 '12 at 20:06

• The point is that no general Eulerian method can resolve phase without resolving CFL ($c \Delta t \sim \Delta x$ where $c$ is the wave speed). If you want to exceed CFL, you have to give up accuracy for some components. If all your waves travel the same speed, you can't gain much. If you have stiff waves or dispersive waves, an implicit method can be an advantage. So the answer is problem-dependent, but your model problem does not offer much potential for implicit methods. – Jed Brown Jan 16 '12 at 15:49