I am trying to solve the 2D/3D wave equation using FEM (c = 1 for simplicity). I have a constraint that the solution should only be in terms of displacement. so
$$\frac{\partial^2 \textbf u}{dt^2}-\Delta \textbf u =f $$
So the sane way to do this would be to introduce a variable to lower the order of the derivative. I am challenged with directly substituting the 2nd order time derivative. I am using Rothe's method so I will not be using a finite difference in the spatial derivatives.
Now I want to try to use Crank Nicolson Scheme but does that exist for a 2nd order finite difference in time? I have seen it with spatial derivatives but nothing with 2nd order time derivatives. I could stick with a plain old backward Euler approach and just plug along.
Any advice here?