Numerical solution of the incompressible Navier-Stokes equations with exponential type schemes

The incompressible Navier-Stokes equations describe a wide range of problems in fluid mechanics. They are composed of an equation in mass conservation and two momentum conservation equations, one for each Cartesian velocity component. The dependent variables will be the pressure p and the velocity components u and v in the x and y directions respectively. The use of numerical methods to solve the governing equations will follow the same methods as used in the previous discussion on numerical solution of non-linear equations. The three governing equations in non-dimensional conservative form are,