Structure-Preserving Numerical Schemes for Nonlinear Adsorption Models with Mass Conservation

Introduction: Nonlinear adsorption models with a conserved total amount are often solved numerically before parameter estimation or model comparison. However, a numerical approximation can be misleading if it does not respect the admissible domain defined by non-negativity and mass conservation. This work studies structure-preserving time discretisation for mass-conserved adsorption dynamics from an applied-mathematics viewpoint.

Methods: We consider a generic batch adsorption model written as dq/dt = F(q,s;theta), ds/dt = -F(q,s;theta), with q(t) denoting the adsorbed phase, s(t) the remaining solute phase, and s(t) + q(t) = m. The conservation law is used to reduce the dynamics to a scalar problem on the interval 0 <= q <= m. Standard one-step discretisations are compared with reduced-variable and projected formulations, using conservation, positivity, and equilibrium admissibility as structural criteria.

Results: The reduced formulation preserves the linear invariant exactly at the discrete level because s is reconstructed as m - q after each update. We also derive an admissibility condition for explicit updates that keeps q_{n+1} inside [0,m], and we identify the situations in which projection is needed to prevent nonphysical iterates. These results separate mass-conservation preservation, which can be imposed algebraically, from positivity preservation, which depends on the flux and the time step.

Conclusions: Structure-preserving discretisation provides a compact framework for simulating nonlinear adsorption kinetics without introducing numerical artefacts that may affect later sensitivity analysis, parameter estimation, or model selection.