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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:

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

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?

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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
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1 Answer

up vote 5 down vote accepted

There is not much point using an implicit method for pure wave propagation because you have to resolve phase to have an accurate method. If you have a hyperbolic system in which some waves are very stiff (not interesting except for their influence on evolution of a slow manifold), you might want an implicit method. It is fairly problem-dependent whether you want an A- or L-stable method, a strong stability preserving method, or a symplectic method. Unfortunately, the order conditions are mutually exclusive such that it is not possible to have all attractive properties in one integration scheme. You can also consider IMEX methods.

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if you want to increase your timestep while preserving or decreasing your spatial resolution, wouldn't that also motivate the need to use an implicit method? –  Paul Jan 16 '12 at 15:30
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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
    
Note that for linear standing wave problems (with sources that have few frequency components), you do not need a time discretization. Instead, you can solve the time-periodic problem directly, although implicit solvers become challenging at high frequency, see scicomp.stackexchange.com/questions/40/… and scicomp.stackexchange.com/questions/619/…. –  Jed Brown Jan 16 '12 at 15:53
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