Finite Volume Methods

  • Oscar Castro-OrgazEmail author
  • Willi H. Hager


The one-dimensional shallow water equations (SWE), or Saint-Venant equations, are a system of nonlinear hyperbolic conservations laws (Toro, Shock-capturing methods for free surface shallow flows. Wiley, Singapore, 2001). The mathematical meaning behind these “surnames” linked to the development of Saint-Venant is clearly elucidated by the definitions (Karni, Lecture notes on numerical methods for hyperbolic equations: short book course. Taylor and Francis, London, 2011; Vazquez-Cendón, Solving hyperbolic equations with finite. Springer, New York, 2015).


  1. Aureli, F., Maranzoni, A., Mignosa, P., & Ziveri, C. (2008). A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography. Advances in Water Resources, 31(7), 962–974.CrossRefGoogle Scholar
  2. Bermudez, A., & Vazquez-Cendón, M. E. (1994). Upwind methods for hyperbolic conservation laws with source terms. Computers & Fluids, 23(8), 1049–1071.MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bradford, S. F., & Sanders, B. F. (2002). Finite-volume model for shallow-water flooding of arbitrary topography. Journal of Hydraulic Engineering, 128(3), 289–298.Google Scholar
  4. Bradford, S. F., & Sanders, B. F. (2005). Performance of high-resolution, nonlevel bed, shallow-water models. Journal of Engineering Mechanics, 131(10), 1073–1081.CrossRefGoogle Scholar
  5. Brocchini, M., & Dodd, N. (2008). Nonlinear shallow water equation modeling for coastal engineering. Journal of Waterway, Port, Coastal, and Ocean Engineering, 134(2), 104–120.Google Scholar
  6. Brufau, P., Vázquez‐Cendón, M. E., & García‐Navarro, P. (2002). A numerical model for the flooding and drying of irregular domains. International Journal for Numerical Methods in Fluids, 39(3), 247–275.Google Scholar
  7. Castro-Orgaz, O., & Chanson, H. (2016). Minimum specific energy and transcritical flow in unsteady open channel flow. Journal of Irrigation and Drainage Engineering, 142(1), 04015030.CrossRefGoogle Scholar
  8. Castro-Orgaz, O., & Chanson, H. (2017). Ritter’s dry-bed dam-break flows: Positive and negative wave dynamics. Environmental Fluid Mechanics, 17(4), 665–694.CrossRefGoogle Scholar
  9. Chaudhry, M. H. (2008). Open-channel flow (2nd ed.). New York: Springer.CrossRefzbMATHGoogle Scholar
  10. Colella, P., & Woodward, P. (1984). The piecewise-parabolic method (PPM) for gas dynamical simulations. Journal of Computational Physics, 54(1), 174–201.zbMATHCrossRefGoogle Scholar
  11. Cunge, J. A. (1975). Rapidly varying flow in power and pumping canals. In K. Mahmood & V. Yevjevich (Eds.), Unsteady flow in open channels, 14, 539–586. Fort Collins, CO, USA: Water Resources Publications.Google Scholar
  12. Cunge, J. A., Holly, F. M., & Verwey, A. (1980). Practical aspects of computational river hydraulics. London: Pitman.Google Scholar
  13. Dressler, R. (1954). Comparison of theories and experiments for the hydraulic dam-break wave. In Proceedings of International of Association of Scientific Hydrology, 3(38), 319–328. Rome, Italy: Assemblée Générale.Google Scholar
  14. Favre, H. (1935). Etude théorique et expérimentale des ondes de translation dans les canaux découverts [Theoretical and experimental study of travelling surges in open channels]. Dunod, Paris, France (in French).Google Scholar
  15. Gharangik, A. M., & Chaudhry, M. H. (1991). Numerical simulation of hydraulic jump. Journal of Hydraulic Engineering, 117(9), 1195–1211.CrossRefGoogle Scholar
  16. Glaister, P. (1987). Difference schemes for the shallow water equations (Numerical Analysis Report 9/87). UK: Department of Mathematics, University of Reading.Google Scholar
  17. Godunov, S. K. (1959). A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Matematicheskii Sbornik, 47(3), 271–306 (in Russian).MathSciNetzbMATHGoogle Scholar
  18. Gottlieb, S., & Shu, C. W. (1998). Total variation diminishing Runge-Kutta schemes. Mathematics of Computation, 67(221), 73–85.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Hafsteinsson, H. J., Evers, F. M., & Hager, W. H. (2017). Solitary wave run-up: wave breaking and bore propagation. Journal of Hydraulic Research, 55(6), 787–798.CrossRefGoogle Scholar
  20. Hancock, S. L. (1980). PISCES industrial simulation code manual. Physics International.Google Scholar
  21. Harten, A. (1983). High resolution schemes for hyperbolic conservation laws. Journal of Computational Physics, 49(3), 357–393.MathSciNetzbMATHCrossRefGoogle Scholar
  22. Harten, A., Lax, P., & van Leer, B. (1983). On upstream differencing and Godunov-type scheme for hyperbolic conservation laws. Society for Industrial and Applied Mathematics, 25(1), 35–61.MathSciNetzbMATHGoogle Scholar
  23. Henderson, F. M. (1966). Open channel flow. New York: MacMillan Co.Google Scholar
  24. Hirsch, C. (1988). Numerical computation of internal and external flows, 1: Fundamentals of numerical discretization. Chichester, England: Wiley.zbMATHGoogle Scholar
  25. Hirsch, C. (1990). Numerical computation of internal and external flows, 2: Computational methods for inviscid and viscous flows. Chichester, England: Wiley.zbMATHGoogle Scholar
  26. Hoffman, J. D. (2001). Numerical methods for engineers and scientists (2nd ed.). New York: Marcel Dekker.zbMATHGoogle Scholar
  27. Jain, S. C. (2001). Open channel flow. New York: Wiley.Google Scholar
  28. Karni, S. (2011). Nonlinear hyperbolic conservation laws-a brief informal introduction. Lecture notes on numerical methods for hyperbolic equations: short book course. London: Taylor and Francis.Google Scholar
  29. Khan, A. A., & Lai, W. (2014). Modeling shallow water flows using the discontinuous Galerkin method. Taylor and Francis, New York: CRC Press.zbMATHCrossRefGoogle Scholar
  30. Lauber, G. (1997). Experimente zur Talsperrenbruchwelle im glatten geneigten Rechteckkanal [Experiments to the dam break wave in smooth sloping rectangular channel]. Ph.D. thesis, ETH Zurich, Zürich, Switzerland (in German).Google Scholar
  31. Lax, P. (1954). Weak solutions of nonlinear hyperbolic equations and their numerical computation. Communications on Pure and Applied Mathematics, 7, 159–193.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Lax, P. D., & Wendroff, B. (1960). Systems of conservation laws. Communications in Pure and Applied Mathematics, 13(2), 217–237.zbMATHCrossRefGoogle Scholar
  33. LeVeque, R. J. (1992). Numerical methods for conservation laws. Basel, Switzerland: Birkhäuser.zbMATHCrossRefGoogle Scholar
  34. LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems. New York: Cambridge University Press.Google Scholar
  35. Mingham, C. G., & Causon, D. M. (1998). High-resolution finite-volume method for shallow water flows. Journal of Hydraulic Engineering, 124(6), 605–614.CrossRefGoogle Scholar
  36. Nujic, M. (1995). Efficient implementation of non-oscillatory schemes for the computation of free-surface flows. Journal of Hydraulic Research, 33(1), 100–111.CrossRefGoogle Scholar
  37. Ozmen-Cagatay, H., & Kocaman, S. (2010). Dam-break flows during initial stage using SWE and RANS approaches. Journal of Hydraulic Research, 48(5), 603–611.CrossRefGoogle Scholar
  38. Ozmen-Cagatay, H., & Kocaman, S. (2011). Dam-break flow in the presence of obstacle: Experiment and CFD simulation. Engineering Applications of Computational Fluid Mechanics, 5(4), 541–552.CrossRefGoogle Scholar
  39. Ritter, A. (1892). Die Fortpflanzung von Wasserwellen [Propagation of water waves]. Zeitschrift Verein Deutscher Ingenieure, 36(2), 947–954 (in German).Google Scholar
  40. Roache, P. J. (1972). Computational fluid dynamics. Albuquerque NM: Hermosa Publishers.zbMATHGoogle Scholar
  41. Roe, P. L. (1981). Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 43(2), 357–372.MathSciNetzbMATHCrossRefGoogle Scholar
  42. Rouse, H. (1961). Fluid motion in a gravitational field. IIHR film, The University of Iowa, Iowa.Google Scholar
  43. Sanders, B. F. (2001). High-resolution and non-oscillatory solution of the St. Venant equations in non-rectangular and non-prismatic channels. Journal of Hydraulic Research, 39(3), 321–330.CrossRefGoogle Scholar
  44. Schoklitsch, A. (1917). Über Dammbruchwellen [On dam break waves]. Kaiserliche Akademie der Wissenschaften, Wien, Mathematisch-Naturwissenschaftliche Klasse. Sitzungberichte IIa, 126, 1489–1514 (in German).zbMATHGoogle Scholar
  45. Sivakumaran, N. S., Tingsanchali, T., & Hosking, R. J. (1983). Steady shallow flow over curved beds. Journal of Fluid Mechanics, 128, 469–487.CrossRefGoogle Scholar
  46. Stoker, J. J. (1957). Water waves: The mathematical theory with applications. New York: Interscience Publishers.zbMATHGoogle Scholar
  47. Sweby, P. K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of the Society for Industrial and Applied Mathematics, Series B Numerical Analysis, 21(5), 995–1011.MathSciNetzbMATHGoogle Scholar
  48. Sweby, P. K. (1999). Godunov methods (Numerical Analysis Report 7/99), UK: Department of Mathematics, University of Reading.Google Scholar
  49. Synolakis, C. E. (1986). The runup of long waves (Ph.D. thesis). California Institute of Technology, Califormia.Google Scholar
  50. Terzidis, G., & Strelkoff, T. (1970). Computation of open channel surges and shocks. Journal of the Hydraulics Division, ASCE, 96(HY12), 2581–2610.Google Scholar
  51. Toro, E. F. (1992). Riemann problems and the WAF method for solving the two-dimensional shallow water equations. Philosophical Transactions of the Royal Society of London, Series A, 338, 43–68.Google Scholar
  52. Toro, E. F. (2001). Shock-capturing methods for free-surface shallow flows. Singapore: Wiley.zbMATHGoogle Scholar
  53. Toro, E. F. (2009). Riemann solvers and numerical methods for fluid dynamics: A practical introduction. Berlin, Germany: Springer.zbMATHCrossRefGoogle Scholar
  54. van Albada, G. D., van Leer, B., & Roberts, W. W. (1982). A comparative study of computational methods in cosmic gas dynamics. Astronomy & Astrophysics, 108(1), 76–84.zbMATHGoogle Scholar
  55. van Leer, B. (1979). Towards the ultimate conservative difference scheme, V: A second-order sequel to Godunov´s method. Journal of Computational Physics, 32, 101–136.zbMATHCrossRefGoogle Scholar
  56. van Leer, B. (2006). Upwind and high-resolution methods for compressible flow: From Donor cell to residual distribution schemes. Computer Physics Communications, 1(2), 192–206.zbMATHGoogle Scholar
  57. Vazquez-Cendón, M. E. (2015). Solving hyperbolic equations with finite volume methods. New York: Springer.zbMATHCrossRefGoogle Scholar
  58. White, F. M. (2009). Fluid mechanics. New York: McGraw-Hill.Google Scholar
  59. Ying, X., Khan, A., & Wang, S. (2004). Upwind conservative scheme for the Saint Venant equations. Journal of Hydraulic Engineering, 130(10), 977–987.CrossRefGoogle Scholar
  60. Ying, X., & Wang, S. (2008). Improved implementation of the HLL approximate Riemann solver for one-dimensional open channel flows. Journal of Hydraulic Research, 46(1), 21–34.MathSciNetCrossRefGoogle Scholar
  61. Zhou, J. G., Causon, D. M., Mingham, C. G., & Ingram, D. M. (2001). The surface gradient method for the treatment of source terms in the shallow water equations. Journal of Computational Physics, 168(1), 1–25.MathSciNetzbMATHCrossRefGoogle Scholar
  62. Zoppou, C., & Roberts, S. (2003). Explicit schemes for dam-break simulations. Journal of Hydraulic Engineering, 129(1), 11–34.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of CórdobaCórdobaSpain
  2. 2.VAW, ETH ZürichZürichSwitzerland

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