Numerical modeling, analysis, and simulations for the space-time wave equation with velocity initial and final time conditions

Two variations of the wave equation are examined in this paper with typical beginning circumstances, the first one match the classical wave equation. The second is a recently suggested space-time vari ational material model where mechanical, thermal, and internal factors interact. It is derived from a thermodynamically consistent Hamilton functional defined across a space time domain. The apparition of a non-standard final time velocity condition in the second order wave equation is a characteristic of this model. We extract and evaluate the wave equation independently, ignoring temperature and internal variables, in order to better grasp the ramifications of the unusual boundary conditions in time. We perform then a Galerkin finite element discretization in time and space for both problem cases. Moreover, we investigate the behavior of these formulations using numerical simulations based on the discretization.