A novel homotopy-wavelet approach for solving stream function-vorticity formulation of Navier-Stokes equations

In this paper, we propose a new homotopy-wavelet approach to solve linear and nonlinear problems with nonhomogeneous boundary conditions. The essence of this technique is to apply the homotopy analysis method (HAM) to transform the governing equations into a set of linear equations and employ the generalized Coiflet-type orthogonal wavelet to express and solve the resulting linear equations. The proposed technique is expected to keep the superiority of the HAM for handling nonlinearities, but with better computational efficiency. The nonhomogeneous boundary conditions including the mixed Dirichlet-Neumann and Robin conditions are reconstructed by introducing the Coiflets on the boundaries, which overcomes the deficiency of the close wavelet method that is difficult to handle the nonhomogeneous boundary conditions. Illustrative examples show very high efficiency of our proposed technique. Furthermore, the classic problem of the incompressible flow in a 2-D lid-driven cavity are investigated. By reconstructing the incompatible boundary conditions with the Coiflets, the singularities of velocity field on rigid points are successfully eliminated so that the vortex on the lid that is difficult to be obtained by previous approaches can be captured clearly. Comparison with previous results is made, excellent agreement is found.