A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics

This paper develops a numerical approach for finding the approximate solution of the Sobolev model. This model describes many natural processes, such as thermal conduction for different media and fluid evolution in soils and rocks. The proposed method approximates the unknown solution with the help of two main stages. At a first stage, the time discretization is performed by means of a second-order finite difference procedure. At a second stage, the space discretization is accomplished using the local radial basis function partition of unity collocation method based on the finite difference (LRBF-PUM-FD). The major disadvantage of global techniques is the high computational burden of solving large linear systems. The LRBF-PUM-FD significantly sparsifies the linear system and reduces the computational burden, while simultaneously maintaining a high accuracy level. The time-discrete formulation is studied in terms of the stability and convergence analysis via the energy method. Three examples are illustrated to verify the efficiency and accuracy of the method.