A new meshfree method of lines for the 3D advection-diffusion equation for solute transport in porous media

Abstract

The study proposes a novel Meshfree Method of Lines (MFMOL) for the numerical solution of three-dimensional advection-diffusion equations (ADEs) used to model solute transport in porous media. The method involves discretising the spatial variables of the model equations using an Augmented Radial Basis Point Interpolation Method (ARPIM), while the temporal variable remains continuous. This results in a system of ordinary differential equations (ODEs), which is numerically integrated using the MATLAB ODE solver. In solving solute transport in porous media, traditional mesh-based methods such as the Finite Element Method (FEM), Finite Difference Method (FDM), and Finite Volume Method (FVM) often encounter challenges such as numerical instabilities and distorted meshes caused by complex geometries and the presence of convective terms. To ensure accuracy, various stabilisation techniques are required, which increase computational effort, cost, and setup time. To overcome these challenges, the new MFMOL approach was proposed and applied in strong-form formulations without stabilisation techniques to solve 3D advection-diffusion problems in homogeneous and isotropic porous media. The results obtained were in good agreement with existing exact solutions, establishing the accuracy and efficiency of the new method and demonstrating its advantages over traditional mesh-based methods for solving problems involving complex geometries.