Semi-analytical solutions of a contaminant transport equation with nonlinear sorption in 1D

Abstract

A 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 solved 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 the Courant-Friedrichs-Lewy (CFL) condition failed). Mathematical modelThe main goal of this paper is to construct precise semi-analytical solutions of the nonlinear convectiondiffusion problem with adsorptionand boundary conditionsHere,where one can assume that δ = 1. The mathematical models (1)-(3) can represent contaminant transport in equilibrium mode with the sorption isotherm (u) = F(u) − u (see, e.g., [9]). The corresponding mathematical model isMost common forms of (nonlinear) sorption isotherms are Freundlichand Langmuirbut one can also consider isotherms of mixed type280 Comput Geosci (2006) 10: 279-290Equation (1) can be written in the form,where the function F (u) ≥ 1 can be viewed as a (nonlinear) "retardation" factor of the convective and diffusive transport.In the case of F (0) = ∞ [i.e., for (4) and (6) with 0 < p < 1], the solution of (1) can have a sharp front with a finite speed of propagation [27]. Moreover, the convective term can be dominant and thus the creation of shocks can be expected even for smooth initial data. Nevertheless, because of the presence of (small) diffusion D 0 > 0, the solution of (1)-(3) is regular and sharp shocks (as known for hyperbolic problems) can not develop in a finite time -see, e.g., [15].Precise numerical solutions of (1) are required, for instance, if one wishes to solve inverse problems (e.g., to determine the diffusion D and the sorption isotherm) or to test numerical methods for this type of problems. Most numerical methods so far are based on regularisation and/or upwinding procedures and they can produce undesirable artefacts in numerical solutions. The main goal of this paper is to avoid any regularisation of F(u) and to significantly decrease numerical dispersion. Consequently, precise numerical solutions of (1)-(3) can be obtained even for the case of vanishing diffusion and for very large retardation factor F (u).Characteristics-based numerical discretisations of contaminant transport with nonlinear adsorption were described in [3,4,8,15], and upwind-based discretization methods for the same type of problems in [6,22]. Asymptotic formulas for large-time behaviour of the exact solution of (1) were obtained in [7, 10], but they can not be used directly in discretization methods.The method presented in this paper is based on operator splitting, where nonlinear transport and nonlinear diffusion are solved separately along each time stepsee, e.g., [12]. The transport part is solved exactl...

Model-based estimation of pesticides and transformation products and their export pathways in a headwater catchment

“…PPDB, 2009). However, the Freundlich isotherm cannot be solved in analytical form (Frolkovič and Kačur, 2006), necessitating numerical approaches. A numerical solution for the calculation of sorption in each time step and cell would have resulted in too long computation times.…”

A global method for coupling transport with chemistry in heterogeneous porous media

“…As this may be too severe a restriction (some of our applications require integration over a very large time interval), an alternative is to use an operator splitting scheme, as proposed by Siegel et al [43] (see also [20,31]). In this work, splitting is used only within the (linear) transport step, but recent papers by Kačur et al [16,22] apply splitting directly to a transport with sorption model by solving (analytically) a nonlinear advection step, followed by a nonlinear diffusion step. This is different from operator splitting as used in geochemical models, as the chemistry terms are solved for together with the transport terms.…”

Numerical modelling of convection-diffusion-reaction problems with free boundary in 1D

“…The conditions (i)-(iii) are also necessary for the existence of the solution of (10). The case 0 < n < 1 is also discussed in [13] (Lemma 13), but we cannot aply it in our analysis.…”

“…17. In this experiment the considered model is very near to nonlinear transport and the solution is very near to the entropy solution of the nonlinear transport with D = 0 -see [10]. To increase the density of the grid points at the front we take M = 20, N = 150.…”

“…But when p → 1 (p = 1 is linear case) troubles arise with our numerical approximations since this is the singular case for our method. The same mathematical model in 1D was treated numerically in [10] using the operator splitting method, where in the transport part a semi-analytical solution has been used and in the diffusion part the finite volume method has been applied. A very precise numerical solution of this model have been applied in a 2D-problem for the transport of contaminant in a dual-well setting.…”