# Mathematical and numerical analysis of a stratigraphic model

• Volume: 38, Issue: 4, page 585-611
• ISSN: 0764-583X

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## Abstract

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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.

## How to cite

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Gervais, Véronique, and Masson, Roland. "Mathematical and numerical analysis of a stratigraphic model." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 38.4 (2004): 585-611. <http://eudml.org/doc/245815>.

@article{Gervais2004,
abstract = {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.},
author = {Gervais, Véronique, Masson, Roland},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {finite volume method; stratigraphic modelling; linear first order equations; convergence analysis; linear advection equation; unique weak solution; adjoint problem},
language = {eng},
number = {4},
pages = {585-611},
publisher = {EDP-Sciences},
title = {Mathematical and numerical analysis of a stratigraphic model},
url = {http://eudml.org/doc/245815},
volume = {38},
year = {2004},
}

TY - JOUR
AU - Gervais, Véronique
AU - Masson, Roland
TI - Mathematical and numerical analysis of a stratigraphic model
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2004
PB - EDP-Sciences
VL - 38
IS - 4
SP - 585
EP - 611
AB - 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.
LA - eng
KW - finite volume method; stratigraphic modelling; linear first order equations; convergence analysis; linear advection equation; unique weak solution; adjoint problem
UR - http://eudml.org/doc/245815
ER -

## References

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1. [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. [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. Zbl0202.36903
3. [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. Zbl1008.74057
4. [4] R. Eymard, T. Gallouët, V. Gervais and R. Masson, Convergence of a numerical scheme for stratigraphic modeling. SIAM J. Numer. Anal. submitted. Zbl1096.35005MR2177876
5. [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. Zbl1098.76618
6. [6] P.B. Flemings and T.E. Jordan, A synthetic stratigraphic model of foreland basin development. J. Geophys. Res. 94 (1989) 3851–3866.
7. [7] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer (1996). Zbl0860.65075MR1410987
8. [8] D. Granjeon, Modélisation Stratigraphique Déterministe: Conception et Applications d’un Modèle Diffusif 3D Multilithologique. Ph.D. Thesis, Géosciences Rennes, Rennes, France (1997).
9. [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. [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. [11] O. Ladyzenskaja, V. Solonnikov and N. Ural’ceva, Linear and quasilinear equations of parabolic type. Transl. Math. Monogr. 23 (1968). Zbl0174.15403
12. [12] J.C. Rivenaes, Application of a dual lithology, depth-dependent diffusion equation in stratigraphic simulation. Basin Research 4 (1992) 133–146.
13. [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. [14] D.M. Tetzlaff and J.W. Harbaugh, Simulating Clastic Sedimentation. Van Norstrand Reinhold, New York (1989).
15. [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.

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