WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms

TL;DR

This paper develops a WENO scheme on characteristics for the equilibrium dispersive model of chromatography with generalized Langmuir isotherms, ensuring accurate, oscillation-free solutions for complex nonlinear PDEs in chromatography modeling.

Contribution

It extends characteristic-based numerical schemes to generalized Langmuir isotherms, proving the well-posedness and eigenstructure of the model, and demonstrates oscillation-free high-resolution solutions.

Findings

The scheme accurately captures steep gradients without oscillations.

The inverse concentration mapping is globally smooth and numerically computable.

Numerical experiments confirm the effectiveness of the method.

Abstract

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modelled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modelling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (Appl. Numer. Math. 123 (2018) 22-42) to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the Jacobian matrix of the system and use it to deduce its eigenstructure. We combine the use of characteristic-based numerical fluxes with a second-order implicit-explicit scheme proposed in the cited reference and perform some numerical experiments with T\'oth's adsorption isotherms to demonstrate that the characteristic-based schemes produce accurate numerical solutions with no oscillations, even when steep gradients appear in the solutions.