On the long-time stability of a temporal discretization scheme for the three dimensional viscous primitive equations

In this article, a semi-discretized Euler scheme to solve the three dimensional viscous primitive equations is studied. Based on suitable assumptions on the initial data and forcing terms, the long-time stability of the proposed scheme is proven by showing that the H¹ norm (in space variables) of the solutions is bounded at each time step when the time step satisfies certain smallness condition.