Nonstandard methods for the convective-dispersive transport equation with nonlinear reactions

Abstract

A new nonstandard Eulerian-Lagrangian method is constructed for the one-dimensional, transient convective-dispersive transport equation with nonlinear reaction terms. An "exact" difference scheme is applied to the convection-reaction part of the equation to produce a semi-discrete approximation with zero local truncation errors with respect to time. The spatial derivatives involved in the remaining dispersion term are then approximated using standard numerical methods. This approach leads to significant, qualitative improvements in the behavior of the numerical solution. It suppresses the numerical instabilities that arise from the incorrect modeling of derivatives and nonlinear reaction terms. Numerical experiments demonstrate the scheme's ability to model convection-dominated, reactive transport problems.

Compatible Discretizations for Continuous Dynamical Systems

“…In [20,21,23,25,26], we experimented with nonstandard methods using the classical nonstandard denominator function:…”

Modeling of Subsurface Biobarrier Formation

“…In Kojouharov and Chen (1999), we proposed a new Eulerian-Lagrangian numerical method for solving the reactive solute transport equation that works very well for Peclet numbers large and small. The numerical solution of the convection-reaction part is defined using an "exact" time-stepping scheme (Kojouharov and Chen, 1998).…”