Localized meshless approaches based on theta method and BDF2 for nonlinear Sobolev equation arising from fluid dynamics

Abstract

This paper formulates the second-order BDF and theta schemes based on an efficient localized meshless technique for computing the approximate solution of the Sobolev model. The model is one of the useful description for modeling the flow of fluids through fractured rocks, movement of moisture in soils. This proposed approach discretizes the nonlinear Sobolev equation in two main steps. In the first step, temporal discretization is accomplished with the help of the second-order BDF and theta schemes, respectively. In the second step, the localized radial basis function partition of unity collocation method (L-RBF-PUCM) is used for spatial discretization, thereby constructing a fully-discrete algorithm. For large linear problems, global techniques have the disadvantage of high computational cost. The proposed method overcomes this shortcoming well and reduces the computational burden by sparsifying the linear system. The convergence analysis and unconditional stability of the semi-discrete formulation are examined by means of the energy method. Numerical results are carried out to verify the validity and accuracy of the L-RBF-PUCM.