# Implicit-Explicit WENO schemes for the equilibrium dispersive model of chromatography

## TL;DR

This paper develops efficient, fully conservative implicit-explicit numerical schemes for the nonlinear equilibrium dispersive model of chromatography, effectively capturing sharp chromatographic fronts in convection-dominated regimes.

## Contribution

It introduces a novel IMEX scheme combining explicit and implicit discretizations for the ED model, leveraging the one-to-one relation between variables for conservation.

## Key findings

- The scheme accurately captures sharp chromatographic fronts.
- It maintains stability similar to hyperbolic problems.
- Numerical experiments demonstrate high accuracy and efficiency.

## Abstract

Chromatographic processes can be modeled by nonlinear, convection-dominated partial differential equations, together with nonlinear relations: the adsorption isotherms. In this paper we consider the nonlinear equilibrium dispersive (ED) model with adsorption isotherms of Langmuir type. We show that very efficient, fully conservative, numerical schemes can be designed for this mode by exploiting the relation between the conserved variables of the model and the physical concentrations of the multi-component mixtures. We show that this relation is one to one and admits a smooth global inverse, which cannot be given explicitly but can be easily computed by using a convenient root finder. These results provide the necessary ingredients to implement fully conservative numerical schemes for the model considered. Implicit-Explicit (IMEX) techniques can be used in the convection-dominated regime in order to increase the efficiency of the numerical scheme. We propose a second order IMEX scheme, combining an explicit Weighted-Essentially-non-Oscillatory discretization of the convective fluxes with an implicit treatment of the diffusive term, in order to illustrate the numerical issues involved in the application of IMEX techniques to this model. Through a series of numerical experiments, we show that the scheme provides accurate numerical solutions which capture the sharp discontinuities present in the chromatographic fronts, with the same stability restrictions as in the purely hyperbolic case.