A space-time accurate method for solving solute transport problems

Solute transport studies frequently rely on numerical solutions of the classical advection-diffusion equation. Unfortunately, solutions obtained with traditional finite difference and finite element techniques typically exhibit spurious damping or oscillation when advection dominates. Recently developed variants of these techniques such as the finite analytic method (Chen and Li, 1979; Chen and Chen, 1984) and the optimal test function method (Celia et al., 1989a, b, c) perform well for steady state problems. Extensions of these methods to the transient case have, however, not been successful, primarily because of inadequate approximations of the temporal derivative. The new numerical method proposed in this paper avoids this difficulty by taking the Laplace transform of the transient equation. The transformed expression behaves like a steady state advection-diffusion equation with a first-order decay term. This expression can be solved with either the finite analytic or optimal test function method and the time dependence recovered with an efficient inverse Laplace transform algorithm. The result is an accurate and robust transient solution which performs well over a very wide range of Peclet numbers. We demonstrate this approach by applying the finite analytic method to a Laplace transformed one-dimensional model problem. A comparison with other competing techniques shows that good approximations are required in both space and time in order to obtain accurate solutions to advection-dominated problems. A good space approximation combined with a poor temporal approximation (or vice versa) does not give satisfactory results. The method we propose provides a balanced space-time approximation which works very well for one-dimensional problems. Extensions to multiple dimensions are conceptually straightforward and briefly discussed.