No abstract

Nonviscous shallow water equations with topography are considered in one spatial dimension. The aim is to find boundary conditions which are physically suitable; that is, they let the waves move freely out of the domain and do not reflect them at the boundary in a nonphysical way (or a way that is believed to be nonphysical). Two types of boundary conditions, called linear and nonlinear, are proposed for two types of flows, namely subcritical and supercritical flows. The second aim in this article is to numerically implement these boundary conditions in a numerically effective way. This is achieved by a suitable extension of the central‐upwind method for the spatial discretization and the Runge‐Kutta method of second order for the time discretization. Several successful numerical experiments for which we tested the proposed boundary conditions and the numerical schemes are described. The topography was added to render the examples studied physically more interesting.

In this article, we prove an asymptotic stability criterion for the solutions of Primitive equations defined on a three-dimensional finite cylindrical domain with time-dependent forcing terms. Under a suitable smallness assumption on the nontrivial forcing terms, we obtain the existence of the time periodic solution for the Primitive equations. Moreover, this time-periodic solution is asymptotically stable in L 2 sense.

In this article, we investigate the time periodic solutions for twodimensional Navier-Stokes equations with nontrivial time periodic force terms. Under the time periodic assumption of the force term, the existence of time periodic solutions for two-dimensional Navier-Stokes equations has received extensive attention from many authors. With the smallness assumption of the time periodic force, we show that there exists only one time periodic solution and this time periodic solution is globally asymptotically stable in the H 1 sense. Without smallness assumption of the force term, there is no stability analysis theory addressed. It is expected that when the amplitude of the force term is increasing, the time periodic solution is no longer asymptotically stable. In the last part of the article, we use numerical experiments to study the bifurcation of the time periodic solutions when the amplitude of the force is increasing. Extrapolating to the heating of the earth by the sun, the bifurcation diagram hints that when the earth receives a relatively small amount of solar energy regularly, the time