A symmetric‐matrix time integration scheme for the efficient solution of advection‐dispersion problems

Abstract

A new time integration scheme for the time‐marching numerical solution of the advection‐dispersion equation is formulated. The scheme leads to a symmetric positive‐definite coefficient matrix that can be solved with any symmetric‐matrix solver. Matrix symmetry is achieved by placing the advective term at the old time level. The resulting numerical errors are minimized by introduction of an artificial diffusion term and by optimal time weighting of all terms on the basis of a Taylor expansion of the governing differential equation. The scheme is unconditionally stable and effectively second‐order accurate, giving results equivalent to those obtained with the Crank‐Nicolson scheme. The use of highly efficient symmetric versions of iterative solution techniques, such as the conjugate gradient method, in the solution of advection‐dispersion problems thus becomes possible. Such techniques can provide substantial advantages in the case of grids with large numbers of nodal points, particularly in three dimensions.