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?