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01.12.2021

A fully well-balanced and asymptotic preserving scheme for the shallow-water equations with a generalized Manning friction source term

verfasst von: Solène Bulteau, Mehdi Badsi, Christophe Berthon, Marianne Bessemoulin-Chatard

Erschienen in: Calcolo | Ausgabe 4/2021

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Abstract

The aim of this paper is to prove the preservation of the diffusive limit by a numerical scheme for the shallow-water equations with a generalized Manning friction source term. This asymptotic behavior coincides with the long time and stiff friction limit. The adopted discretization was initially developed to preserve all the steady states of the model under concern. In this work, a relevant improvement is performed in order to preserve also the diffusive limit of the problem and to exactly capture the moving and non-moving steady solutions. In addition, a second-order time and space extension is detailed. Involving suitable linearizations, the obtained second-order scheme exactly preserves the steady states and the diffusive behavior. Several numerical experiments illustrate the relevance of the designed schemes.

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Alonso, R., Santillana, M., Dawson, C.: On the diffusive wave approximation of the shallow water equations. Eur. J. Appl. Math. 19(5), 575–606 (2008)MathSciNetCrossRef

Andreianov, B., Boyer, F., Hubert, F.: Finite volume schemes for the $p$-Laplacian on Cartesian meshes. M2AN Math. Model. Numer. Anal. 38(6), 931–959 (2004)MathSciNetCrossRef

Andreu, F., Mazón, J.M., Segura de León, S., Toledo, J.: Existence and uniqueness for a degenerate parabolic equation with $L^1$-data. Trans. Am. Math. Soc. 351(1), 285–306 (1999)CrossRef

Argyros, I.K., Ren, H.: On the convergence of modified Newton methods for solving equations containing a non-differentiable term. J. Comput. Appl. Math. 231(2), 897–906 (2009)MathSciNetCrossRef

Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25(6), 2050–2065 (2004)MathSciNetCrossRef

Barrett, J.W., Liu, W.B.: Finite element approximation of the $p$-Laplacian. Math. Comput. 61(204), 523–537 (1993)MathSciNetMATH

Bermudez, A., Vazquez, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23(8), 1049–1071 (1994)MathSciNetCrossRef

Berthon, C., Chalons, C.: A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Math. Comput. 85(299), 1281–1307 (2016)MathSciNetCrossRef

Berthon, C., Charrier, P., Dubroca, B.: An HLLC scheme to solve the $M_1$ model of radiative transfer in two space dimensions. J. Sci. Comput. 31(3), 347–389 (2007)MathSciNetCrossRef

10.

Berthon, C., LeFloch, P.G., Turpault, R.: Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations. Math. Comput. 82(282), 831–860 (2013)MathSciNetCrossRef

11.

Berthon, C., Turpault, R.: Asymptotic preserving HLL schemes. Numer. Methods Partial Differ. Equ. 27(6), 1396–1422 (2011)MathSciNetCrossRef

12.

Buet, C., Cordier, S.: An asymptotic preserving scheme for hydrodynamics radiative transfer models: numerics for radiative transfer. Numer. Math. 108(2), 199–221 (2007)MathSciNetCrossRef

13.

Buet, C., Despres, B.: Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys. 215(2), 717–740 (2006)MathSciNetCrossRef

14.

Castro, M.J., Pardo Milanés, A., Parés, C.: Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique. Math. Models Methods Appl. Sci. 5, 2055–2113 (2007)MathSciNetCrossRef

15.

Coquel, F., Godlewski, E.: Asymptotic preserving scheme for Euler system with large friction. J. Sci. Comput. 48(1–3), 164–172 (2011)MathSciNetCrossRef

16.

Díaz, J., Martínez, A.L.: Movimiento de descarga de gases en conductos largos: modelizacion y estudio de una ecuacion parabolica doblemente no lineal. In: Actas de la Reunión Matemática en Honor de A. Dou, Madrid, Universidad Complutense de Madrid (1989)

17.

Diaz, J.I., De Thelin, F.: On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25(4), 1085–1111 (1994)MathSciNetCrossRef

18.

Díaz, J.I., Herrero, M.A.: Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems. Proc. R. Soc. Edinb. Sect. A 89(3–4), 249–258 (1981)MathSciNetCrossRef

19.

Dubroca, B., Feugeas, J.-L.: Étude théorique et numérique d’une hiérarchie de modèles aux moments pour le transfert radiatif. C. R. Acad. Sci. Paris Sér. I Math. 329(10), 915–920 (1999)

20.

Duran, A., Marche, F., Berthon, C., Turpault, R.: Numerical discretizations for shallow water equations with source terms on unstructured meshes. In: Hyperbolic Problems: Theory, Numerics, Applications, volume 8 of AIMS Ser. Appl. Math., pp. 541–549. Am. Inst. Math. Sci. (AIMS), Springfield, MO (2014)

21.

Duran, A., Marche, F., Turpault, R., Berthon, C.: Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes. J. Comput. Phys. 287, 184–206 (2015)MathSciNetCrossRef

22.

Gosse, L.: A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39(9–10), 135–159 (2000)MathSciNetCrossRef

23.

Gosse, L., Toscani, G.: An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R. Math. Acad. Sci. Paris 334(4), 337–342 (2002)MathSciNetCrossRef

24.

Greenberg, J.M., Leroux, A.Y.: A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33(1), 1–16 (1996)MathSciNetCrossRef

25.

Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)MathSciNetCrossRef

26.

Herrero, M.A., Vazquez, J.L.: Asymptotic behaviour of the solutions of a strongly nonlinear parabolic problem. Ann. Fac. Sci. Toulouse Math. Ser. 5 3(2), 113–127 (1981)MathSciNetCrossRef

27.

Hsiao, L., Liu, T.-P.: Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 143(3), 599–605 (1992)MathSciNetCrossRef

28.

Jin, S.: Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21(2), 441–454 (1999)MathSciNetCrossRef

29.

Ju, N.: Numerical analysis of parabolic $p$-Laplacian: approximation of trajectories. SIAM J. Numer. Anal. 37(6), 1861–1884 (2000)MathSciNetCrossRef

30.

Kamin, S., Vazquez, J.L.: Fundamental solutions and asymptotic behaviour for the $p$-laplacian equation (1988)

31.

LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge (2002)CrossRef

32.

Manning, R.: On the flow of water in open channels and pipes. Trans. Inst. Civ. Eng. Irel. 20, 161–207 (1890)

33.

Marcati, P., Milani, A.: The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Differ. Equ. 84(1), 129–147 (1990)MathSciNetCrossRef

34.

Marche, F., Berthon, C., Turpault, R.: An efficient scheme on wet/dry transitions for shallow water equations with friction. Comput. Fluids 48(1), 192–201 (2011)MathSciNetCrossRef

35.

Michel-Dansac, V.: Développement de schémas équilibre d’ordre élevé pour des écoulements géophysiques. PhD thesis, Université de Nantes (2016)

36.

Michel-Dansac, V., Berthon, C., Clain, S., Foucher, F.: A well-balanced scheme for the shallow-water equations with topography or Manning friction. J. Comput. Phys. 335, 115–154 (2017)MathSciNetCrossRef

37.

Noelle, S., Xing, Y., Shu, C.-W.: High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226(1), 29–58 (2007)MathSciNetCrossRef

38.

Osher, S.: Convergence of Generalized Muscl Schemes, pp. 134–148. Springer, Berlin, Heidelberg (1997)

39.

Peton, N.: Étude et simulation d’un modèle stratigraphique advecto-diffusif non-linéaire avec frontières mobiles. PhD thesis, Université Paris-Saclay (2018)

40.

Segura de León, S., Toledo, J.: Regularity for entropy solutions of parabolic $p$-Laplacian type equations. Publ. Mat. 43(2), 665–683 (1999)MathSciNetCrossRef

41.

Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, 3rd edn. Springer, Berlin (2009)CrossRef

42.

van Leer, B.: Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32(1), 101–136 (1979)CrossRef

43.

Xu, J., Kawashima, S.: Diffusive relaxation limit of classical solutions to the damped compressible Euler equations. J. Differ. Equ. 256(2), 771–796 (2014)MathSciNetCrossRef

Titel: A fully well-balanced and asymptotic preserving scheme for the shallow-water equations with a generalized Manning friction source term
verfasst von: Solène Bulteau
Mehdi Badsi
Christophe Berthon
Marianne Bessemoulin-Chatard
Publikationsdatum: 01.12.2021
Verlag: Springer International Publishing
Erschienen in: Calcolo / Ausgabe 4/2021
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-021-00432-7

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A fully well-balanced and asymptotic preserving scheme for the shallow-water equations with a generalized Manning friction source term

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