Complex solution to nonideal contaminant transport through porous media

Nonideal or nonequilibrium transport through porous media is described by a convection-diffusion equation coupled to a first order kinetics accounting for mass transfer between the solid and the fluid phases. The overall mathematical model may be formulated using an integro-differential approach and very effectively Laplace transformed with complex parameters p_k, k=0,1,...,2M+1. Solution in Laplace space may be addressed by finite elements (FE). The resulting complex valued FE equations can be solved with either a complex or an equivalent real arithmetic operating on a problem twice as large as the original one. For both approaches preconditioned projection (or conjugate gradient-like) methods are used. Particularly difficult problems with high Peclet numbers are investigated as well. The results from three representative test cases totaling up to 15,000 equations show that the complex solution approach is superior to the real approach by up to almost two orders of magnitude, depending on problem. It is also shown that while the solver performance vs p_k is stable in complex arithmetic, this does not hold true for the solver in real arithmetic, and an argument based on the quality of preconditioning is offered to account for the observed different computational behavior.