I would like to solve $$\alpha u_{tt} = -\nabla^2u$$ with $\frac{\partial u}{\partial n} = 1$. On using a Galerkin approximation I obtain $$M\ddot{c}=\frac{1}{\alpha}(Dc+b)$$ where $M$ is the mass matrix and and $D$ is the stiffness matrix. I note that I can write this in the form $$\dot{U} = ZU+B$$
where $U = [c, \dot{c}]^T$ and $Z$ has off diagonal blocks top right: $I$ and bottom left $M^{-1}D$ and $B$ has only non-trivial entries in the "second half" of the vector given by $M^{-1}b$.
We can go ahead and use in implicit finite difference scheme on this problem then.(?)
I know my explanation is sparse but I am hoping that someone who might have solved this before can confirm whether this is a viable method. My solution is very "unstable" looking.
Perhaps I am being really naive using this method. Are there simple methods to solve this problem?
