Finite volume approximations of shear shallow water model on unstructured grids
Tiwari, Shashwat1 ; Nkonga, Boniface2 ; Chandrashekar, Praveen1 ; Gavrilyuk, Sergey3
1Center for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore – 560065, India.
2Université Côte d’Azur, INRIA, CNRS, LJAD, 06108 Nice Cedex 2, France.
3Aix Marseille Univ., CNRS, IUSTI, Marseille.
The SMAI Journal of computational mathematics, Tome 10 (2024), pp. 229-261

We develop and implement a finite volume method based on an almost conservative approach for the hyperbolic non-conservative system of the shear shallow water model. This model is a generalization of the shallow water model to include the effects of shear, which are ignored in the standard shallow water model. The existing path conservative scheme is extended to unstructured grids composed of triangles or quadrilaterals with the solution stored at the cell centers. The high-resolution scheme is based on a local solution reconstruction idea for which a new set of variables based on a Cholesky decomposition of the Reynolds stress tensor (fluctuation of horizontal velocity) is developed, which is found to increase the robustness of the schemes by maintaining positive definiteness property. The scheme is applied to some test problems like 1D dam break, square dam break, roll waves, and radial hydraulic jump.

Publié le :
DOI : 10.5802/smai-jcm.112
Classification : 65M08, 76M12
Keywords: shallow flow, shear flow, non-conservative system

Tiwari, Shashwat  1   ; Nkonga, Boniface  2   ; Chandrashekar, Praveen  1   ; Gavrilyuk, Sergey  3

1 Center for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore – 560065, India.
2 Université Côte d’Azur, INRIA, CNRS, LJAD, 06108 Nice Cedex 2, France.
3 Aix Marseille Univ., CNRS, IUSTI, Marseille. 
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     author = {Tiwari, Shashwat and Nkonga, Boniface and Chandrashekar, Praveen and Gavrilyuk, Sergey},
     title = {Finite volume approximations of shear shallow water model on unstructured grids},
     journal = {The SMAI Journal of computational mathematics},
     pages = {229--261},
     year = {2024},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {10},
     doi = {10.5802/smai-jcm.112},
     zbl = {07963390},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/smai-jcm.112/}
}
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Tiwari, Shashwat; Nkonga, Boniface; Chandrashekar, Praveen; Gavrilyuk, Sergey. Finite volume approximations of shear shallow water model on unstructured grids. The SMAI Journal of computational mathematics, Tome 10 (2024), pp. 229-261. doi: 10.5802/smai-jcm.112

[1] Abgrall, Rémi; Karni, Smadar A comment on the computation of non-conservative products, J. Comput. Phys., Volume 229 (2010) no. 8, pp. 2759-2763 (Accessed 2020-02-27) | MR | DOI | Zbl

[2] Balay, Satish; Abhyankar, Shrirang; Adams, Mark F.; Brown, Jed; Brune, Peter; Buschelman, Kris; Dalcin, Lisandro; Dener, Alp; Eijkhout, Victor; Gropp, William D.; Kaushik, Dinesh; Knepley, Matthew G.; May, Dave A.; McInnes, Lois Curfman; Mills, Richard Tran; Munson, Todd; Rupp, Karl; Sanan, Patrick; Smith, Barry F.; Zampini, Stefano; Zhang, Hong; Zhang, Hong PETSc Users Manual (2018) no. ANL-95/11 - Revision 3.10 http://www.mcs.anl.gov/petsc (Technical report)

[3] Barth, Timothy J. Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier–Stokes Equations, AGARD, Special Course on Unstructured Grid (1992), 19920018434 (Accessed 2022-07-25)

[4] Berthon, Christophe Numerical approximations of the 10-moment Gaussian closure, Math. Comput., Volume 75 (2006) no. 256, pp. 1809-1831 http://www.ams.org/... (Accessed 2019-03-22) | MR | DOI | Zbl

[5] Bhole, Ashish; Nkonga, Boniface; Gavrilyuk, Sergey; Ivanova, Kseniya Fluctuation splitting Riemann solver for a non-conservative modeling of shear shallow water flow, J. Comput. Phys., Volume 392 (2019), pp. 205-226 https://linkinghub.elsevier.com/... (Accessed 2019-07-09) | MR | DOI | Zbl

[6] Brock, R. The development of roll-waves trains in open channels, J. Hydraulics Division, Volume 95 (1969), pp. 1401-1428 | DOI

[7] Brock, R. Periodic permanent roll waves, J. Hydraulics Division, Volume 96 (1970), pp. 2565-2580 | DOI

[8] Bush, John W. M.; Aristoff, Jeffrey M.; Hosoi, A. E. An experimental investigation of the stability of the circular hydraulic jump, J. Fluid Mech., Volume 558 (2006), pp. 33-52 | DOI | Zbl

[9] Busto, Saray; Dumbser, Michael; Gavrilyuk, Sergey; Ivanova, Kseniya On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows, J. Sci. Comput., Volume 88 (2021) no. 1, 28, 45 pages https://link.springer.com/... (Accessed 2021-08-17) | MR | DOI | Zbl

[10] Castro, M. J.; Fernández-Nieto, E. D.; Ferreiro, A. M.; García-Rodríguez, J. A.; Parés, C. High Order Extensions of Roe Schemes for Two-Dimensional Nonconservative Hyperbolic Systems, J. Sci. Comput., Volume 39 (2009) no. 1, pp. 67-114 http://link.springer.com/10.1007/s10915-008-9250-4 (Accessed 2024-07-01 bibtex: Castro2009) | MR | Zbl | DOI

[11] Chalons, Christophe; Coquel, Frédéric A new comment on the computation of non-conservative products using Roe-type path conservative schemes, J. Comput. Phys., Volume 335 (2017), pp. 592-604 https://linkinghub.elsevier.com/... (Accessed 2024-07-01 bibtex: Chalons2017) | MR | DOI | Zbl

[12] Chandrashekar, Praveen; Nkonga, Boniface; Meena, Asha Kumari; Bhole, Ashish A path conservative finite volume method for a shear shallow water model, J. Comput. Phys., Volume 413 (2020), 109457, 28 pages https://linkinghub.elsevier.com/... (Accessed 2020-07-08) | MR | DOI | Zbl

[13] Dal Maso, Gianni; Lefloch, Philippe G.; Murat, Francois Definition and weak stability of nonconservative products, J. Math. Pures Appl., Volume 74 (1995), pp. 483-548 | MR | Zbl

[14] Foglizzo, Thierry; Masset, Frédéric; Guilet, Jérôme; Durand, Gilles Shallow Water Analogue of the Standing Accretion Shock Instability: Experimental Demonstration and a Two-Dimensional Model, Phys. Rev. Lett., Volume 108 (2012) no. 5, 051103 (Accessed 2021-06-28) | DOI

[15] Gallice, Gerard; Chan, Agnes; Loubere, Raphael; Maire, Pierre-Henri Entropy stable and positivity preserving Godunov-type schemes for multidimensional hyperbolic systems on unstructured grid, J. Comput. Phys., Volume 468 (2022), 111493, 33 pages | MR | DOI | Zbl

[16] Gavrilyuk, Sergey; Ivanova, Kseniya; Favrie, Nicolas Multi-dimensional shear shallow water flows: Problems and solutions, J. Comput. Phys., Volume 366 (2018), pp. 252-280 https://linkinghub.elsevier.com/... (Accessed 2019-03-22) | MR | DOI | Zbl

[17] Geuzaine, Christophe; Remacle, Jean-François Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Methods Eng., Volume 79 (2009) no. 11, pp. 1309-1331 (Accessed 2018-01-01) | MR | DOI | Zbl

[18] Ivanova, Kseniya; Gavrilyuk, Sergey Structure of the hydraulic jump in convergent radial flows, J. Fluid Mech., Volume 860 (2019), pp. 441-464 https://www.cambridge.org/... (Accessed 2021-06-28) | MR | DOI | Zbl

[19] Lax, Peter; Wendroff, Burton Systems of conservation laws, Commun. Pure Appl. Math., Volume 13 (1960) no. 2, pp. 217-237 (Accessed 2021-08-09 bibtex: Lax1960) | MR | DOI | Zbl

[20] Levermore, C. David; Morokoff, William J. The Gaussian Moment Closure for Gas Dynamics, SIAM J. Appl. Math., Volume 59 (1998) no. 1, pp. 72-96 (Accessed 2019-03-22) | MR | DOI | Zbl

[21] McLachlan, Robert I.; Quispel, G. Reinout W. Splitting methods, Acta Numer., Volume 11 (2002), pp. 341-434 http://www.journals.cambridge.org/... (Accessed 2018-09-15) | MR | DOI | Zbl

[22] Nkonga, Boniface; Chandrashekar, Praveen Exact solution for Riemann problems of the shear shallow water model, ESAIM, Math. Model. Numer. Anal., Volume 56 (2022) no. 4, pp. 1115-1150 | MR | DOI | Zbl

[23] Parés, Carlos Numerical methods for nonconservative hyperbolic systems: a theoretical framework., SIAM J. Numer. Anal., Volume 44 (2006) no. 1, pp. 300-321 (Accessed 2019-03-24) | MR | DOI | Zbl

[24] Richard, Gäel; Gavrilyuk, Sergey The classical hydraulic jump in a model of shear shallow-water flows, J. Fluid Mech., Volume 725 (2013), pp. 492-521 | MR | DOI | Zbl

[25] Strang, Gilbert On the Construction and Comparison of Difference Schemes, SIAM J. Numer. Anal., Volume 5 (1968) no. 3, pp. 506-517 (Accessed 2018-09-15) | MR | DOI | Zbl

[26] Tadmor, Eitan Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., Volume 12 (2003), pp. 451-512 http://www.journals.cambridge.org/... (Accessed 2019-05-07) | MR | DOI | Zbl

[27] Teshukov, Vladimir M. Gas-dynamic analogy for vortex free-boundary flows, J. Appl. Mech. Tech. Phys., Volume 48 (2007) no. 3, pp. 303-309 http://link.springer.com/10.1007/s10808-007-0039-2 (Accessed 2019-03-22) | DOI

[28] Yadav, Anshu; Kumar, Harish; Chandrashekar, Praveen; Bhoriya, Deepak Entropy stable schemes for the shear shallow water model Equations, J. Sci. Comput., Volume 97 (2023) no. 3, 77, 36 pages | MR | Zbl

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