I am trying to model the seismic wave equation and have therefore been reading about discretization schemes and their stability. I recently came across an insightful paper on 'Galerkin FEM methods for wave equation' http://www.math.tamu.edu/~bangerth/publications/2010-waves.pdf. In this document and many other documents, for the newmark beta scheme, the final discretised form of wave equation is in terms of acceleration, velocity and displacement. That is how I have seen it in other documents as well.
However, I have also seen another form of Newmark scheme of wave equation, i.e., the following: (Pg 130)
Now, I am really confused as to how a wave equation can have 2 different forms of Newmark discretisation. I' ll be very grateful if you could shed some light on this or point me in the right direction.
The end purpose is to model it in fenics and DEAL 2