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Mathematical and numerical analysis of a stratigraphic model. (English) Zbl 1130.86315

Summary: 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 \(c_i^s\) 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.

MSC:

86A60 Geological problems
35L50 Initial-boundary value problems for first-order hyperbolic systems
35L65 Hyperbolic conservation laws
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
74L05 Geophysical solid mechanics
86-08 Computational methods for problems pertaining to geophysics
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References:

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