Mathematical and numerical analysis of a stratigraphic model
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, p. 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.

DOI : https://doi.org/10.1051/m2an:2004035
Classification:  35M10,  35L50,  35Q99,  65M12
Keywords: finite volume method, stratigraphic modelling, linear first order equations, convergence analysis, linear advection equation, unique weak solution, adjoint problem
@article{M2AN_2004__38_4_585_0,
     author = {Gervais, V\'eronique and Masson, Roland},
     title = {Mathematical and numerical analysis of a stratigraphic model},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {4},
     year = {2004},
     pages = {585-611},
     doi = {10.1051/m2an:2004035},
     zbl = {1130.86315},
     mrnumber = {2087725},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2004__38_4_585_0}
}
Gervais, Véronique; Masson, Roland. Mathematical and numerical analysis of a stratigraphic model. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, pp. 585-611. doi : 10.1051/m2an:2004035. http://www.numdam.org/item/M2AN_2004__38_4_585_0/

[1] R.S. Anderson and N.F. Humphrey, Interaction of Weathering and Transport Processes in the Evolution of Arid Landscapes, in Quantitative Dynamics Stratigraphy, T.A. Cross Ed., Prentice Hall (1989) 349-361.

[2] C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels ; théorèmes d'approximation ; application à l'équation de transport. Ann. Sci. École Norm. Sup. 3 (1971) 185-233. | Numdam | Zbl 0202.36903

[3] A. Blouza, H. Le Dret, An up-to-the boundary version of Friedrichs' lemma and applications to the linear Koiter shell model. SIAM J. Math. Anal. 33 (2001) 877-895. | Zbl 1008.74057

[4] R. Eymard, T. Gallouët, V. Gervais and R. Masson, Convergence of a numerical scheme for stratigraphic modeling. SIAM J. Numer. Anal. submitted. | MR 2177876 | Zbl 1096.35005

[5] R. Eymard, T. Gallouët, D. Granjeon, R. Masson and Q.H. Tran, Multi-lithology stratigraphic model under maximum erosion rate constraint. Int. J. Numer. Meth. Eng. 60 (2004) 527-548. | Zbl 1098.76618

[6] P.B. Flemings and T.E. Jordan, A synthetic stratigraphic model of foreland basin development. J. Geophys. Res. 94 (1989) 3851-3866.

[7] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996). | MR 1410987 | Zbl 0860.65075

[8] D. Granjeon, Modélisation Stratigraphique Déterministe: Conception et Applications d'un Modèle Diffusif 3D Multilithologique. Ph.D. Thesis, Géosciences Rennes, Rennes, France (1997).

[9] D. Granjeon and P. Joseph, Concepts and applications of a 3D multiple lithology, diffusive model in stratigraphic modeling, in J.W. Harbaugh et al. Eds., Numerical Experiments in Stratigraphy, SEPM Sp. Publ. 62 (1999).

[10] P.M. Kenyon and D.L. Turcotte, Morphology of a delta prograding by bulk sediment transport, Geological Society of America Bulletin 96 (1985) 1457-1465.

[11] O. Ladyzenskaja, V. Solonnikov and N. Ural'Ceva, Linear and quasilinear equations of parabolic type. Transl. Math. Monogr. 23 (1968). | Zbl 0174.15403

[12] J.C. Rivenaes, Application of a dual lithology, depth-dependent diffusion equation in stratigraphic simulation. Basin Research 4 (1992) 133-146.

[13] J.C. Rivenaes, Impact of sediment transport efficiency on large-scale sequence architecture: results from stratigraphic computer simulation. Basin Research 9 (1997) 91-105.

[14] D.M. Tetzlaff and J.W. Harbaugh, Simulating Clastic Sedimentation. Van Norstrand Reinhold, New York (1989).

[15] G.E. Tucker and R.L. Slingerland, Erosional dynamics, flexural isostasy, and long-lived escarpments: A numerical modeling study. J. Geophys. Res. 99 (1994) 229-243.