Numerical Solution of Two-Dimensional Advection Diffusion Equation for Multiphase Flows in Porous Media Using a Novel Meshfree Method of Lines. - Kazeem, Jamiu Abiodun; Ogunfiditimi, Franklin Olusodayo (International Journal of Applied Sciences and Smart Technologies, 2025)
This study proposes a novel Meshfree Method of Lines (MFMOL), in strong form formulation, to solve multiphase flows of solute transport modelled by two-dimensional (2D) Advection Diffusion Equations (ADE). The method uses a consistent and stable Augmented Radial Basis Point Interpolation Method (ARPIM) for spatial variables discretization of the models, while the time variable is left continuous, resulting in a system of Ordinary Differential Equations (ODEs) with initial conditions, which is solved numerically, via Matlab ode solver. The new method is proposed to overcome the challenges of numerical instabilities and large deformation due to complex domain, and distorted or low-quality meshes that attracts remeshing, all encountered by traditional Finite Element Method (FEM), Finite Difference Method (FDM) and Finite Volume Method (FVM). Also, the MFMOL is used in strong form formulation without any stabilization techniques for the convective terms in solute transport models, contrary to other methods like FEM, FDM, FVM and meshfree Finite Point Method (FPM) that require the stabilization techniques for fluid flow problems to guarantee acceptable results. The efficiency and accuracy of the new method were established and validated by using it to solve 2D diffusive and advective flow problems in the complex domain of porous structures. The results obtained agreed with the existing exact solutions, using less computational efforts, costs and time, compared with mesh-based methods and others that require stabilization for the convective terms. These features established the superior performance of the new method to the mesh-based methods and others that require special stabilization techniques for solving 2D multiphase flow of solute transport in porous media and other transient fluid flow problems.