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Cited by 108Computational Fluid Dynamics and AerodynamicsAquatic and Environmental StudiesNonlinear Waves and Solitons

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Finite-volume schemes for shallow-water equations

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Kurganov, Alexander·2018·Acta Numerica
DOI 10.1017/s0962492918000028
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Abstract

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.

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  1. 10.1007/s10915-012-9578-7 · DOI
  2. 10.1007/s10915-010-9377-y · DOI
  3. 10.1016/j.advwatres.2009.11.004 · DOI
  4. 10.1061/(ASCE)0733-950X(2000)126:3(144) · DOI
  5. 10.1016/0021-9991(79)90145-1 · DOI
  6. 10.1006/jcph.2001.6892 · DOI
  7. 10.1016/j.jcp.2006.01.044 · DOI
  8. Hundsdorfer, 2007, IMEX extensions of linear multistep methods with general monotonicity and boundedness properties, J. Comput. Phys., 225, 2016, 10.1016/j.jcp.2007.03.003 · DOI
  9. 10.1016/j.jcp.2007.08.007 · DOI
  10. 10.1006/jcph.1994.1187 · DOI
  11. M. Mignotte and D. Stefanescu (2002), ‘On an estimation of polynomial roots by Lagrange’. Technical report 025/2002, IRMA Strasbourg. http://hal.archives-ouvertes.fr/hal-00129675/en/
  12. 10.1142/9789814417099_0061 · DOI
  13. 10.1051/m2an:2008019 · DOI
  14. Cockburn, 1998, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, 10.1007/BFb0096351 · DOI
  15. Bollermann, 2011, Finite volume evolution Galerkin methods for the shallow water equations with dry beds, Commun. Comput. Phys., 10, 371, 10.4208/cicp.220210.020710a · DOI
  16. Bouchut, 2010, A robust well-balanced scheme for multi-layer shallow water equations, Discrete Contin. Dyn. Syst. Ser. B, 13, 739, 10.3934/dcdsb.2010.13.739 · DOI
  17. Vabishchevich, 2014, Additive Operator-Difference Schemes: Splitting Schemes
  18. Manning, 1891, On the flow of water in open channel and pipes, Transactions of the Institution of Civil Engineers of Ireland, 20, 161
  19. Gauckler, 1867, Etudes Théoriques et Pratiques sur l’Ecoulement et le Mouvement des Eaux
  20. 10.1016/j.jcp.2012.01.007 · DOI
  21. 10.1006/jcph.1996.0130 · DOI
  22. 10.1137/S1064827500373413 · DOI
  23. Pareschi, 2005, Implicit–Explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., 25, 129
  24. 10.1137/0721062 · DOI
  25. Richtmyer, 1994, Difference Methods for Initial-Value Problems
  26. 10.1007/s11433-008-0135-1 · DOI
  27. 10.1002/(SICI)1097-0363(19991215)31:7<1037::AID-FLD909>3.0.CO;2-V · DOI
  28. 10.1016/j.jcp.2011.03.042 · DOI
  29. 10.1007/88-470-0354-7_18 · DOI
  30. 10.1029/2010WR009751 · DOI
  31. Castro Díaz, 2014, Solving two-mode shallow water equations using finite volume methods, Commun. Comput. Phys., 16, 1323, 10.4208/cicp.180513.230514a · DOI
  32. 10.1002/fld.1355 · DOI
  33. X. Liu , J. Albright , Y. Epshteyn and A. Kurganov (2018), ‘Well-balanced positivity preserving central-upwind scheme with a novel wet/dry reconstruction on triangular grids for the Saint-Venant system’. Submitted. · DOI
  34. Harten, 1993, Algorithmic Trends in Computational Fluid Dynamics, 287, 10.1007/978-1-4612-2708-3_17 · DOI
  35. M. J. Castro Díaz , A. Kurganov and T. Morales de Luna (2018), ‘Path-conservative central-upwind schemes for nonconservative hyperbolic systems’. Preprint. · DOI
  36. 10.1002/cpa.3160070112 · DOI
  37. 10.1002/num.20049 · DOI
  38. 10.1137/080719091 · DOI
  39. 10.1137/0705041 · DOI
  40. 10.1016/j.compfluid.2011.04.011 · DOI
  41. 10.1016/j.jhydrol.2009.12.020 · DOI
  42. 10.1007/b79761 · DOI
  43. Lie, 2003a, An improved quadrature rule for the flux-computation in staggered central difference schemes in multidimensions, J. Sci. Comput., 63, 1539
  44. 10.1137/06067167X · DOI
  45. 10.1017/CBO9780511791253 · DOI
  46. Russo, 2010, WASCOM 2009: 15th Conference on Waves and Stability in Continuous Media, 230
  47. Hu, 2012, Numerical modelling of turbidity currents in the Xiaolangdi reservoir, Yellow River, China, J. Hydrol., 464, 41, 10.1016/j.jhydrol.2012.06.032 · DOI
  48. 10.1016/j.jcp.2008.10.020 · DOI
  49. Schijf, 1953, Minnesota International Hydraulics Convention, 321
  50. Arminjon, 1995, Généralisation du schéma de Nessyahu–Tadmor pour une équation hyperbolique à deux dimensions d’espace, CR Acad. Sci. Paris Sér. I Math., 320, 85
  51. 10.1007/978-1-4612-0713-9 · DOI
  52. Kellerhals, 1967, Stable channels with gravel-paved beds, ASCE Journal of the Waterways and Harbors Division, 93, 63, 10.1061/JWHEAU.0000482 · DOI
  53. 10.1016/j.jcp.2014.09.027 · DOI
  54. Chertock, 2018, Well-balanced schemes for the Euler equations with gravitation: Conservative formulation using global fluxes, J. Comput. Phys., 358, 36, 10.1016/j.jcp.2017.12.026 · DOI
  55. 10.1017/CBO9780511546785 · DOI
  56. 10.1137/S1064827599359461 · DOI
  57. 10.1016/j.jcp.2008.05.012 · DOI
  58. Pareschi, 2001, Recent Trends in Numerical Analysis, 269
  59. 10.1080/10618569908940819 · DOI
  60. Flamant, 1891, Mécanique Appliquée: Hydraulique
  61. 10.1137/S1064827503431090 · DOI
  62. 10.1061/(ASCE)0733-9429(2007)133:1(48) · DOI
  63. Marchuk, 1990, Finite Difference Methods, I: Solution of Equations in ℝ N, 197
  64. 10.1016/j.advwatres.2011.04.017 · DOI
  65. 10.1051/m2an:2002019 · DOI
  66. 10.1137/070686147 · DOI
  67. 10.1137/070679065 · DOI
  68. 10.1137/0907051 · DOI
  69. Chertock, 2009, Numerical Methods for Balance Laws, 303
  70. 10.1137/S1064827501385852 · DOI
  71. 10.1080/02626660109492804 · DOI
  72. 10.1007/s10092-001-8181-3 · DOI
  73. Kurganov, 2017, Second-order fully discrete central-upwind scheme for two-dimensional hyperbolic systems of conservation laws, SIAM J. Sci. Comput., 39, A947, 10.1137/15M1038670 · DOI
  74. 10.4310/CMS.2007.v5.n1.a6 · DOI
  75. 10.1137/S106482759631041X · DOI
  76. 10.1007/s002110050345 · DOI
  77. 10.1016/0021-9991(90)90260-8 · DOI
  78. 10.1051/m2an:2001108 · DOI
  79. 10.1080/1061856031000104851 · DOI
  80. Rusanov, 1961, The calculation of the interaction of non-stationary shock waves with barriers, Ž. Vyčisl. Mat. i Mat. Fiz., 1, 267
  81. Godunov, 1959, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, Mat. Sb. (N.S.), 47, 271
  82. 10.1137/040614189 · DOI
  83. 10.1137/151005798 · DOI
  84. de Saint-Venant, 1871, Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lits, CR Acad. Sci. Paris, 73, 147
  85. 10.1006/jcph.1994.1148 · DOI
  86. 10.1080/10618569808940837 · DOI
  87. 10.1016/S0168-9274(97)00056-1 · DOI
  88. Exner, 1920, Zur Physik der Dünen
  89. Beljadid, 2016, Well-balanced positivity preserving cell-vertex central-upwind scheme for shallow water flows, Comput. and Fluids, 136, 193, 10.1016/j.compfluid.2016.06.005 · DOI
  90. 10.1016/0021-9991(89)90065-X · DOI
  91. 10.1007/978-3-540-75712-2_33 · DOI
  92. 10.1016/0021-9991(84)90013-5 · DOI
  93. A. Bernstein , A. Chertock , A. Kurganov and T. Wu (2018), ‘Central-upwind scheme for shallow water equations with time-dependent bottom topography’. Preprint.
  94. 10.1137/S1064827597324998 · DOI
  95. 10.1051/m2an/2010060 · DOI
  96. 10.1137/040607642 · DOI
  97. 10.1051/m2an:2004041 · DOI
  98. 10.1016/S1007-5704(03)00115-1 · DOI
  99. 10.1090/S0025-5718-09-02317-5 · DOI
  100. 10.1142/S021820250700256X · DOI
  101. Cheng, 2016, Moving-water equilibria preserving central-upwind schemes for the shallow water equations, Commun. Math. Sci., 14, 1643, 10.4310/CMS.2016.v14.n6.a9 · DOI
  102. Y. Cheng , A. Chertock , M. Herty , A. Kurganov and T. Wu (2018), ‘A new approach for designing moving-water equilibria preserving schemes for the shallow water equations’. Preprint. · DOI
  103. 10.1002/fld.4023 · DOI
  104. 10.1007/s10915-010-9381-2 · DOI
  105. Chertock, 2005, Finite Volumes for Complex Applications IV, 63
  106. 10.3846/13926292.2013.869269 · DOI
  107. 10.1016/0021-9991(87)90031-3 · DOI
  108. Dal Maso, 1995, Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), 74, 483
  109. Darcy, 1857, Recherches Expérimentales Relatives au Mouvement de l’Eau dans les Tuyaux, 1
  110. 10.1016/j.cma.2013.01.006 · DOI
  111. 10.1016/j.cma.2013.09.022 · DOI
  112. 10.1002/cpa.3160070206 · DOI
  113. 10.1016/S0045-7930(02)00011-7 · DOI
  114. Gerbeau, 2001, Derivation of viscous Saint-Venant system for laminar shallow water: Numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 1, 89, 10.3934/dcdsb.2001.1.89 · DOI
  115. 10.1142/7498 · DOI
  116. Gottlieb, 2001, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 89, 10.1137/S003614450036757X · DOI
  117. Guermond, 2016, Fast estimation from above of the maximum wave speed in the Riemann problem for the Euler equations, J. Comput. Phys., 321, 908, 10.1016/j.jcp.2016.05.054 · DOI
  118. 10.1016/0021-9991(81)90128-5 · DOI
  119. 10.1137/0724022 · DOI
  120. 10.1137/1025002 · DOI
  121. 10.1061/(ASCE)0733-950X(1992)118:3(249) · DOI
  122. Higueras, 2006, Ninth International Conference Zaragoza–Pau on Applied Mathematics and Statistics, 129
  123. 10.1016/j.cam.2014.05.011 · DOI
  124. 10.1016/0021-9991(89)90226-X · DOI
  125. 10.1016/j.mcm.2010.09.005 · DOI
  126. 10.1137/S0036142997317560 · DOI
  127. 10.1051/m2an:2001130 · DOI
  128. Kröner, 1997, Numerical Schemes for Conservation Laws
  129. Kurganov, 2006, European Conference on Computational Fluid Dynamics: ECCOMAS CFD 2006
  130. Kurganov, 2016, Handbook of Numerical Methods for Hyperbolic Problems, 525
  131. 10.1137/S1064827599360236 · DOI
  132. Kurganov, 2007, On the reduction of numerical dissipation in central-upwind schemes, Commun. Comput. Phys., 2, 141
  133. Kurganov, 2014, Central-upwind scheme for Savage–Hutter type model of submarine landslides and generated tsunami waves, Comput. Methods Appl. Math., 14, 177, 10.1515/cmam-2014-0003 · DOI
  134. 10.1002/nme.1471 · DOI
  135. 10.1137/100795280 · DOI
  136. 10.1007/PL00005455 · DOI
  137. 10.1006/jcph.2000.6459 · DOI
  138. 10.1006/jcph.1998.6058 · DOI
  139. Bernstein, 2016, Central-upwind scheme for shallow water equations with discontinuous bottom topography, Bull. Braz. Math. Soc. (N.S.), 47, 91, 10.1007/s00574-016-0124-3 · DOI
  140. 10.1002/num.10025 · DOI
  141. 10.1007/978-3-0348-8150-0 · DOI
  142. 10.1142/S0219891604000287 · DOI
  143. 10.1051/m2an:1999152 · DOI
  144. 10.1007/s10915-012-9677-5 · DOI
  145. 10.1137/S1064827501392880 · DOI
  146. 10.1137/0733038 · DOI
  147. 10.1137/S1064827502419439 · DOI
  148. 10.1016/j.jcp.2006.06.015 · DOI
  149. 10.4208/cicp.2009.v6.p848 · DOI
  150. 10.1007/s10915-010-9425-7 · DOI
  151. 10.1016/j.jcp.2010.08.006 · DOI
  152. 10.1016/j.jcp.2005.08.019 · DOI
  153. 10.1016/j.jcp.2007.03.031 · DOI
  154. Parés, 2009, Numerical Methods for Balance Laws, 67
  155. 10.1017/S0022112002002550 · DOI
  156. 10.1016/j.compfluid.2015.11.017 · DOI
  157. Shu, 1988, Total-variation-diminishing time discretizations, SIAM J. Sci. Comput., 6, 1073, 10.1137/0909073 · DOI
  158. 10.1016/0021-9991(88)90177-5 · DOI
  159. 10.1016/j.cageo.2006.02.020 · DOI
  160. 10.1007/s00033-003-1119-7 · DOI

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Metadata

Authors
Kurganov, A.
Affiliations
Tulane University (US)
Corresponding author
Kurganov, Alexander
ORCID
Kurganov, Alexander
Venue
Acta Numerica
Publisher
Cambridge University Press (CUP)
Published
2018-05-01
Volume
27
Pages
289-351
ISSN
  • 0962-4929 (ISSN-L)
  • 1474-0508
DOI
10.1017/s0962492918000028
OpenAlex
W2799904623
Cited by
108
References
0
Citations by year
2026: 9 · 2025: 18 · 2024: 15 · 2023: 17 · 2022: 18 · 2021: 13 · 2020: 15 · 2019: 2 · 2018: 1
FWCI
5.58
Type
journal article
Language
en
Open access
Licence
www.cambridge.org/core/terms (unspecified, 3 d)
Locations
Acta Numerica (publishedVersion)
Topics
  • Physical Sciences › Engineering › Computational Mechanics › Computational Fluid Dynamics and Aerodynamics
  • Physical Sciences › Earth and Planetary Sciences › Earth-Surface Processes › Aquatic and Environmental Studies
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Nonlinear Waves and Solitons
Keywords
shallow water equations, Saint-Venant system, finite volume method, central-upwind schemes, Godunov-type schemes, Riemann solvers, numerical diffusion, friction term, non-conservative hyperbolic systems, non-smooth solutions

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