5: Iterative schemes and convergence analysis

Abstract

Iterative approaches are frequently used to solve intricate coupled and highly nonlinear systems. As stated in the previous chapters the mathematical model that governed the nanoparticles transport in porous media consists of equations of pressure, nanoparticles concentration, deposited nanoparticles concentration on the pore-walls, and entrapped nanoparticles concentration in pore-throats. In the case of the two-phase flow, the saturation equation is also considered. Iterative methods are often employed to solve such kinds of time-dependent complicated systems. The nonlinear iterative numerical scheme such as iterative Implicit Pressure Concentration, and the iterative Implicit Pressure Explicit Saturation–Implicit Concentration (IMPES-IMC) has been introduced to solve the model under consideration. The iterative IMP-IMC scheme is devoted to solving the problem of nanoparticles transport with single-phase flow in porous media, while the iterative IMPES-IMC treats the two-phase flow case. This chapter focuses on the iterative methods and investigates their theoretical and numerical convergence. Moreover, a theoretical foundation for the convergence of the iterative approach has been proved using the mathematical induction method.