Numerical Methods for Solving Some Hydrodynamic Stability Problems

In this paperwe solve the stability problem of the thermal convection in a thin fluid layer with free-free, slip-slip, and fixed-slip boundary conditions. We use different numerical methods to check their flexibility and accuracy where we use the following numerical methods: finite difference, High order finite difference, p order finite element, Chebyshev collocation-1, Chebyshev collocation-2 and Chebyshev tau. Moreover, we solve the equations of Poiseuille flow in a Brinkman porous media with slip-slip boundary conditions using p order finite element, Chebyshev collocation-1 and Chebyshev collocation-2 methods. The advantage and disadvantage of each method have been discussed. According to our result, we believe that the finite difference and finite element methods are very flexible methods and we can apply them to solve any problem easily. However, the accuracy of the the finite difference and finite element methods is less than the accuracy of the Chebyshev methods.