The Cell Analytical‐Numerical Method for Solution of the Advection‐Dispersion Equation: Two‐Dimensional Problems

Abstract

In this paper we present the development and evaluation of a new numerical scheme for efficient solution of groundwater solute transport problems. The scheme, which we have named the cell analytical‐numerical (CAN) method, is an extension of the so‐called nodal methods which have been developed in the nuclear engineering area for solving neutron diffusion problems. The CAN method is based upon decomposition of the solution domain into a number of rectangular cells (volume subdomains); each cell is homogeneous so that a local analytic solution to the solute transport equation can be obtained. A first‐order accurate finite difference approximation of the time derivative followed by a transverse averaging procedure is used to transform the governing partial differential equation into a set of coupled one‐dimensional ordinary differential equations for the transverse moments of the concentration. The local analytical solution is therefore found in terms of these moments. Solute mass flux continuity across cell surfaces, along with the local analytical solution, is used to construct an algebraic relationship between concentration moment values at adjacent cell surfaces. Assembling all the cells together results in a set of coupled tridiagonal matrix equations, one set for each spatial direction, which can be solved very efficiently. The CAN method is applied to several simple test problems involving uniform flow in homogeneous aquifers. While the method lacks geometrical flexibility, it offers a spatial approximation that is demonstrated to have high accuracy and minimal grid orientation error, even when applied to coarse meshes.