Mathematical and numerical analysis of a stratigraphic model
Institut Français du Pétrole, 1 et 4 avenue de Bois Préau, 92852 Rueil Malmaison Cedex, France. email@example.com.
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 in lithology i of the sediments at the top of the basin, and the L concentrations ci 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 with a linear advection equation for ci for which 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.
Mathematics Subject Classification: 35M10 / 35L50 / 35Q99 / 65M12
Key words: Finite volume method / stratigraphic modelling / linear first order equations / convergence analysis / linear advection equation / unique weak solution / adjoint problem.
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