Non-Standard Methods for the Convective-Dispersive Transport Equation
with Nonlinear Reactions
Abstract
Convection-dispersion-reaction equations arise in many
chemical
and biological settings. In groundwater aquifers, they govern kinetic
adsorption and the growth and transport of biofilm-forming microbes.
Accurate simulation in these cases is crucial to the development of
contaminant remediation strategies. However, convection interferes with
the numerics, causing spurious oscillations or artificial diffusion
near sharp fronts.
In contrast with linear reactions, nonlinear reactions may contain
thresholds, and errors near these thresholds can affect the qualitative
structure of solutions. Therefore, both the oscillations associated
with poorly resolved, nondiffusive schemes and the smearing associated
with diffusive schemes can distort the numerical solutions
significantly.
We examine a new class of Eulerian-Lagrangian numerical methods
constructed according to non-standard modeling rules. The
convection-reaction
equation, obtained by ignoring the dispersion term, is approximated
using
an "exact" difference scheme. Having dealt with the most difficult part
of the problem, numerically; standard finite differences and finite
elements are well suited for the remaining dispersion term.
Some of the important features of the proposed new methods are the
nonlocal treatment of nonlinear reaction terms and the use of more
complicated discretization of time derivatives. This approach leads to
significant, qualitative improvements in the behavior of the numerical
solutions. As theory predicts, the usual numerical difficulties are
absent or greatly reduced. Large time steps can be taken without
affecting
the accuracy of the numerical solutions.
