A comparative study of time integration schemes for groundwater flow and transport using the differential quadrature method of lines

The accurate modeling of groundwater flow and contaminant transport is essential for managing water resources and ensuring environmental safety, especially in arid and semi-arid regions where groundwater often constitutes the primary freshwater supply. This study presents a comprehensive numerical analysis of various time integration schemes applied in conjunction with the Differential Quadrature Method (DQM) for solving one- and two-dimensional groundwater flow and transport problems. Spatial derivatives are approximated using DQM based on Chebyshev-Gauss-Lobatto (CGL) collocation points, while the temporal domain is discretized via different schemes including Explicit Euler (EE), Implicit Euler (IE), Crank-Nicolson (CN), classical Runge-Kutta (RK4), and MATLAB’s adaptive solvers (ode45 and ode15s). The governing partial differential equations (PDEs) are transformed into systems of ordinary differential equations (ODEs) using the method of lines (MOL), enabling a comparative analysis of these time integration strategies. In addition to standard benchmark problems, the study evaluates numerical performance on high-resolution grids and heterogeneous aquifers with full dispersion tensors. The results demonstrate that while classical fixed-step schemes encounter stability limitations at high spatial resolutions, the combination of DQM with adaptive solvers maintains robustness and accuracy. Comparisons indicate that the proposed framework yields lower error margins than the standard FDM using significantly fewer grid nodes. Furthermore, the explicit ode45 solver is found to be efficient for smooth regimes, whereas the implicit ode15s solver is shown to be effective for stiff problems. The findings highlight the practical suitability of these adaptive numerical strategies for simulating complex groundwater phenomena.