ADI method based on C2-continuous two-node integrated-RBF elements for viscous flows

We propose a C2-continuous alternating direction implicit (ADI) method for the solution of the streamfunction-vorticity equations governing steady 2D incompressible viscous fluid flows. Discretisation is simply achieved with Cartesian grids. Local two-node integrated radial basis function elements (IRBFEs) [D.-A. An-Vo, N. Mai-Duy, T. Tran-Cong, A C2-continuous control-volume technique based on Cartesian grids and two-node integrated-RBF elements for second-order elliptic problems, CMES: Comput. Model. Eng. Sci. 72 (2011) 299-334] are used for the discretisation of the diffusion terms, and then the convection terms are incorporated into system matrices by treating nodal derivatives as unknowns. ADI procedure is applied for the time integration. Following ADI factorisation, the two-dimensional problem becomes a sequence of one-dimensional problems. The solution strategy consists of multiple use of a one-dimensional sparse matrix algorithm that helps saving the computational cost. High levels of accuracy and efficiency of the present methods are demonstrated with solutions of several benchmark problems defined on rectangular and non-rectangular domains.