Mathematical and numerical analysis of a stratigraphic model

Gervais, Véronique; Masson, Roland

ESAIM: Modélisation mathématique et analyse numérique, Tome 38 (2004) no. 4, pp. 585-611.

In this paper, we consider a multi-lithology diffusion model used in stratigraphic modelling to simulate large scale transport processes of sediments described as a mixture of $L$ lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main unknowns of the system are the sediment thickness $h$, the $L$ surface concentrations $c_i^s$ in lithology $i$ of the sediments at the top of the basin, and the $L$ concentrations $c_i$ in lithology $i$ of the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for $c_i^s$ with a linear advection equation for $c_i$ for which $c_i^s$ appears as an input boundary condition. For this coupled system, a weak formulation is introduced which is shown to have a unique solution. An implicit finite volume scheme is derived for which we show stability estimates and the convergence to the weak solution of the problem.

Classification : 35M10, 35L50, 35Q99, 65M12
Mots clés : finite volume method, stratigraphic modelling, linear first order equations, convergence analysis, linear advection equation, unique weak solution, adjoint problem

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