Semi-analytical solutions for contaminant transport with nonlinear sorption in 1D

New method to determine semi-analytical solutions of one-dimensional contaminant transport problem with nonlinear sorption is described. This method is based on operator splitting approach where the convective transport is realized exactly and the diffusive transport by finite volume method. The exact solutions for all sorption isotherms of Freundlich and Langmuir type are presented for the case of piecewise constant initial profile and zero diffusion. Very precise numerical results for transport with small diffusion can be obtained even for larger time steps (e.g. when CFL condition failed). 1 Mathematical model The main goal of this paper is to construct precise semi-analytical solutions for nonlinear convection-diffusion problem with adsorption ∂tF (u) + v(x)∂xu − ∂x(D(x, t)∂xu) = f(x, u) (1) for x ∈ (0, L), t > 0 and with boundary conditions u(0, t) = C(t), ∂xu(L, t) = 0 (2) and initial condition u(x, 0) = u0(x). (3) Here v(x) ≥ v0 > 0, D(x, t) ≥ D0 > 0, F (0) = 0, F (s) ≥ δ, (4) where one can assume that δ = 1. ∗This work was funded by the Federal Ministry of Economics and Technology (BMWi) under the contract number 02 E 9148 2 and the first author was also partially supported by grants MSM 260100001 and GARC 201/00/0557