A stable and accurate compact exponential scheme for 3D groundwater pollution model: Theory and Comparison

This paper presents and compares two fully discrete high-order difference schemes for solving three-dimensional convection–diffusion equation relevant to groundwater modeling. The first is the proposed compact exponential finite difference scheme and the second is a standard compact finite difference scheme. It is important to note that the standard compact finite difference scheme often fails to accurately capture sharp gradients and boundary layers. To overcome these challenges, we propose a compact exponential finite difference scheme, which effectively handles both limitations while offering enhanced numerical accuracy and stability. In both approaches, the alternating direction implicit method is employed to enhance computational efficiency. The spatial derivatives in the first scheme are approximated using compact exponential operators, while the second uses classical fourth-order compact finite difference operators. Theoretical analysis of the proposed compact exponential scheme is carried out in the H1-norm, which shows that the method is unconditionally stable and achieves a convergence rate of order O(Δt2+Δx4+Δy4+Δz4). Numerical experiments, including a realistic groundwater pollution scenario, validate the theoretical findings and demonstrate that our proposed compact exponential scheme offers improved accuracy, particularly in convection-dominated problems and near boundary layers, compared to the standard compact finite difference scheme.