Abstract
Finite volume (FV) methods for solving the two-dimensional (2D) nonlinear shallow water equations (NSWE) with source terms on unstructured, mostly triangular, meshes are known for some time now. There are mainly two basic formulations of the FV method: node-centered (NCFV) and cell-centered (CCFV). In the NCFV formulation the finite volumes, used to satisfy the integral form of the equations, are elements of the mesh dual to the computational mesh, while for the CCFV approach the finite volumes are the mesh elements themselves. For both formulations, details are given of the development and application of a second-order well-balanced Godunov-type scheme, developed for the simulation of unsteady 2D flows over arbitrary topography with wetting and drying. The popular approximate Riemann solver of Roe is utilized to compute the numerical fluxes, while second-order spatial accuracy is achieved with a MUSCL-type reconstruction technique. The Green-Gauss (G-G) formulation for gradient computations is implemented for both formulations, in order to maintain a common framework. Two different stencils for the G-G gradient computations in the CCFV formulation are implemented and tested. An edge-based limiting procedure is applied for the control of the total variation of the reconstructed field. This limiting procedure is proved to be effective for the NCFV scheme but inadequate for the CCFV approach. As such, a simple but very effective modification to the reconstruction procedure is introduced that takes into account geometrical characteristics of the computational mesh. In addition, consistent well-balanced second-order discretizations for the topography source term treatment and the wet/dry front treatment are presented for both FV formulations, ensuring absolute mass conservation, along with a stable friction term treatment.
Using a controlled environment for a fair comparison, a complete assessment of both FV formulations is attempted through rigorous individual and relative performance comparisons to the approximation of analytical benchmark solutions, as well as to experimental and field data. To this end, an extensive evaluation is performed using different time dependent and steady-state test cases that incorporate topography, wetting and drying process, different types of boundary conditions as well as friction. These test cases are chosen as to compare the performance and robustness of each formulation under certain conditions and evaluate the effectiveness of the proposed modifications. Emphasis to grid convergence studies is given, with the grids used to range from regular grids to irregular ones with random perturbations of nodes. The results indicate that the quality of the mesh has a major impact on the convergence performance of the CCFV method while the NCFV method exhibits a uniform behavior on the different grid types used. The proposed correction in the computation of the reconstructed values for the CCFV formulation greatly improves the convergence behavior to the formal order of accuracy.
Similar content being viewed by others
References
Abgrall R (1994) On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation. J Comput Phys 114:45
Aftosmis M, Gaitone D, Tavares TS (1994) On the accuracy, stability and monotonicity of various reconstruction algorithms for unstructure meshes. AIAA paper 94-0415
Aftosmis M, Gaitone D, Tavares TS (1995) Behavior of linear reconstruction techniques on unstructured meshes. AIAA J 33:2038
Alcrudo F, Gil E (1999) The Malpasset dam-break case study. In: Proceeding of the 4rd CADAM workshop, Zaragoza
Aliabadi S, Akbar M, Patel R (2010) Hybrid finite element/volume method for shallow water equations. Int J Numer Methods Eng 83:2719
Ambati VR, Bokhove O (2007) Space-time discontinuous Galerkin finite element method for shallow water flows. J Comput Appl Math 204:452
Audusse E, Bristeau M-O (2005) A well-balanced positivity preserving second order scheme for shallow water flows on unstructured meshes. J Comput Phys 206:311
Aureli F, Maranzoni P, Ziveri C (2008) A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography. Adv Water Resour 31:962
Barth TJ, Ohlberger M (2004) Finite volume methods: foundation and analysis. In: Stein E, de Borst R, Hudges TR (eds) Encyclopedia of computational mechanics. Wiley, New York
Barth TJ (1992) Aspects of unstructured grids and finite volume solvers for the Euler and Navier-Stokes equations. In: Special course on unstructured grid methods for advection dominated flows, AGARD report, p 787
Barth TJ (2003) Numerical methods and error estimation for conservation laws on structured and unstructured meshes. VKI computational fluid dynamics lecture series
Barth TJ, Jespersen DC (1989) The design and application of upwind schemes on unstructured meshes. AIAA paper 89-0366
Batten P, Lambert C, Causon DM (1996) Positively conservative high-resolution convection schemes for unstructured elements. Int J Numer Methods Fluids 39:1821
Baumbach K, Lukác̆ová-Medvidová (2008) On the comparison of evolution Galerkin and discontinuous Galerkin schemes. In: Jorgensen P, Shen X, Shu C-W, Yan N (eds) Recent advances in computational sciences. World Scientific, Singapore
Begnudelli L, Sanders BF (2006) Unstructured grid finite volume algorithm for shallow water flow and scalar transport with wetting and drying. ASCE J Hydraul Eng 132(4):371
Berger M, Aftosmis MJ, Murman SM (2005) Analysis of slope limiters on irregular grids. AIAA paper 2005-0490, 43rd AIAA Aerospace Sciences Meeting and Exhibit: Reno, Nevada
Bermudez A, Dervieux A, Desideri JA, Vázquez ME (1998) Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes. Comput Methods Appl Mech Eng 155(1–2):49
Bermudez A, Vázquez-Cendón ME (1994) Upwind methods for hyperbolic conservation laws with source terms. Comput Fluids 23:1049
Blain CA, Massey TC (2005) Application of a coupled discontinuous Galerkin finite-element finite element shallow water model to coastal ocean dynamics. Ocean Model 10:283
Blazek J (2006) Computational fluid dynamics. Elsevier, Amsterdam
Bokhove O (2005) Flooding and drying in discontinuous Galerkin finite-element discretizations of shallow-water equations. Part 1: One dimension. J Sci Comput 22(3):47
Bouchut F (2004) Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhauser, Basel
Bova SW, Carey GF (1996) An entropy variable formulation and applications for the two-dimensional shallow water equations. Int J Numer Methods Fluids 23:26
Bradford SF, Sanders BF (2002) Finite-volume model for shallow-water flooding of arbitary torography. ASCE J Hydraul Eng 128:289
Bristeau M-O, Coussin B (2001) Boundary conditions for the shallow water equations solved by kinetic schemes. Raport de Recherche No 4282, INRIA
Brocchini M, Bernetti R, Mancinelli A, Albertini G (2001) An efficient solver for nearshore flows based on the WAF method. Coast Eng 43:105
Brocchini M, Dodd N (2008) Nonlinear shallow water equations modeling for coastal engineering. J Waterw Port Coastal Ocean Eng 134:104
Brufau P, García-Navarro P, Vázquez-Cendón ME (2004) Zero mass error using unsteady wetting-drying coditions in shallow flows over dry irregular topography. Int J Numer Methods Fluids 45:1047–1082
Brufau P, Vázquez-Cendón ME, Gracía-Navarro P (2002) A numerical model for the flooding and drying of irregular domain. Int J Numer Methods Fluids 39:247
Bryson S, Levy D (2005) Balanced central schemes for the shallow water equations on unstructured grids. SIAM J Sci Comput 27(2):532
Bunya S, Kubatko EJ, Westerink JJ, Dawson C (2009) A wetting and drying treatment for the Runge-Kutta discontinuous Galerkin solution to the shallow water equations. Comput Methods Appl Mech Eng 198:1548
Burguete J, Murillo J, Garcia-Navarro P (2006) Numerical boundary conditions for globally mass conservative methods to solve shallow-water equations and applied to river flow. Int J Numer Methods Fluids 51:585
Caleffi V, Valiani A, Zanni A (2003) Finite volume method for simulating extreme flood events in natural channels. J Hydraul Res 41:167
Canestrelli A, Dumbster M, Siviglia A, Toro EF (2010) Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed. Adv Water Resour 33:291
Castro CE, Kaser M, Toro EF (2009) Space-time adaptive numerical methods for geophysical applications. Philos Trans R Soc Phys Eng Sci 367:4613
Castro MJ, Ferreiro AM, García-Rodriguez JA, González-Vida JM, Macías J, Parés C, Vázquez-Cendón ME (2005) The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems. Math Comput Model 42:419
Castro MJ, González-Vida JM, Parés C (2006) Numerical treatment of wet/dry fronts in shallow flows with a modified Roe scheme. Math Models Methods Appl Sci 16:897
Casulli V, Zanolli P (2007) Comparing analytical and numerical solution of nonlinear two and three-dimensional hydrostatic flows. Int J Numer Methods Fluids 53:1049
Cea L, French JR, Vázquez-Cendón ME (2006) Numerical modelling of total flows in complex estuaries including turbulence: an unstructured finite volume solver and experimental validation. Int J Numer Methods Eng 67:1909
Cea L, Puertas J, Vázquez-Cendón ME (2007) Depth averaged modelling of turbulent shallow water flow with wet-dry fronts. Arch Comput Methods Eng 14:303
Chidaglia J-M (1998) Flux schemes for solving nonlinear systems of conservation laws. In: Chattot JJ, Hafez M (eds) Proceedings of the meeting in honor of P.L. Roe, Arcachon, July 1998
Delis AI, Kazolea M, Kampanis NA (2008) A robust high resolution finite volume scheme for the simulation of long waves over complex domain. Int J Numer Methods Fluids 56:419
Delis AI, P Skeels C (1998) TVD schemes for open channel flow. Int J Numer Methods Fluids 26:791
Diskin B, Thomas J (2008) Accuracy of gradient reconstructions on grids with high aspect ratio. NIA Report 2008-12, National Institute of Aerospace
Diskin B, Thomas JL (2010) Comparison of node-centered and cell-centered unstructured finite volume discretizations: inviscid fluxes. AIAA paper 2010-1079, 48th AIAA Aerospace sciences meeting including the New Horizons Forum and Aerospace Exposition: Orlando, Florida
Diskin B, Thomas JL, Nielsen EJ, Nishikawa H (2009) Comparison of node-centered and cell-centered unstructured finite volume discretizations. Part I: viscous fluxes. AIAA paper 2009-597, 47th AIAA Aerospace sciences meeting including the New Horizons Forum and Aerospace Exposition: Orlando, Florida
Erduran KS, Kutija V, Hewett CJM (2002) Performance of finite volume solutions to the shallow water equations with shock-capturing schemes. Int J Numer Methods Fluids 40:1237
Ern A, Piperno S, Djadel K (2008) A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying. Int J Numer Methods Fluids 58:1
Gallardo JM, Parés C, Castro M (2007) On a well-balanced higher-order finite volume scheme for shallow water equations with topography and dry areas. J Comput Phys 227:574
Gallouet T, Herard J-M, Seguin N (2003) Some approximate Godunov schemes to compute shallow water equations with topography. Comput Fluids 32:479
George DL (2008) Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. J Comput Phys 227:3089
George DL Adaptive finite volume methods with well-balanced Riemann solvers for modeling floods in rugged terrain: application to the Malpasset dam-break flood (France, 1959). Int J Numer Methods Fluids, in press
Godlewski E, Raviart PA (1995) Hyperbolic systems of conservation laws. Applied mathematical sciences, vol 118. Springer, Berlin
Godunov SK (1959) Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat Sb 47:271
Goutal N (1999) The Malpasset dam failure—an overview and test case definition. In: Proceeding of the 4rd CADAM workshop, Zaragoza
Guinot V (2003) Riemann solvers and boundary conditions for two-dimensional shallow water simulations. Int J Numer Methods Fluids 41:1191
Harten A, Hyman P (1983) Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J Comput Phys 50:235
Hervouet J-M (2000) A high-resolution 2-d dam-break model using parallelization. Hydrol Process 14:2211
Hervouet J-M (2007) Hydrodynamics of free surface flows: modelling with the finite element method. Wiley, New York
Hervouet J-M, Petitjean A (1999) Malpasset dam-break revisited with 2-dimensional computations. J Hydraul Res 37:777
Hirsch C (1990) Numerical computation of internal and external flows. Wiley, Chichester
Hubbard ME (1999) Multidimensional slope limiters for MUSCL-type finite volume schemes on unstructured grids. J Comput Phys 155:54
Hubbard ME, Dodd N (2002) A 2D numerical model of wave runup and overtoping. Coast Eng 47:1
Hubbard ME, García-Navarro P (2000) Flux difference splitting and the balancing of source terms and flux gradients. J Comput Phys 165:2
Hunter NM, Bates PD, Neelz S, Pender G, Villanueva I, Wright NG, Liang D, Falconer RA, Lin B, Waller S, Crossley AJ, Mason DC (2008) Benchmarking 2D hydraulic models for urban flood simulations. Proc Inst Civ Eng-Wat Mgmt 161:13
Idelson SR, On̆ate E (1994) Finite volumes and finite elements: Two good friends. Int J Numer Methods Eng 37:3323
Kesserwani G, Liang QH, Vazquez J, Mose R (2010) Well-balancing issues related to the RKDG2 scheme for the shallow water equations. Int J Numer Methods Fluids 62:428
Kubatko EJ, Westerink JJ, Dawson C (2006) hp discontinuous Galerkin methods for advection dominated problems in shallow water flow. Comput Methods Appl Mech Eng 196:437
Kubatko SJ, Bunya E, Dawson C, Westerink C, Mirabito JJ (2009) A performance comparison of continuous and discontinuous finite element shallow water models. J Sci Comput 40:315
Kurganov A, Petrova G (2007) A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system. Commun Math Sci 5:133
Lallemand MH (1988) Etude de schémas Runge-Kutta á 4 pas pour la résolution multigrille des équations d’Euler 2D. Raport de Recherche, INRIA
Lallemand MH (1990) Dissipative properties of Runge-Kutta schemes with upwind spatial approximation for the Euler equations. Raport de Recherche No 1173, INRIA
Leveque RJ (1998) Balancing source terms and flux gradients in high-resolution Godunov-type methods. J Comput Phys 24:346
LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge
LeVeque RJ, George DL (2007) High-resolution finite volume methods for the shallow water equations with bathymetry and dry states. In: Yeh H, Liu PL, Synolakis CE (eds) Advanced numerical models for simulating tsunami waves and runup. Advances in coastal and ocean engineering, vol 10. World Scientific, Singapore
Liang D, Binliang L, Falconer RA (2007) Simulation of rapidly varying flow using an efficient TVD-MacCormack scheme. Int J Numer Methods Fluids 53:811
Liang Q, Borthwick AGL (2009) Adapive quadtree simulation of shallow flows with wet-dry fronts over complex topography. Comput Fluids 38:221
Lukác̆ová-Medvidová M, Teschke U (2006) Comparison study of some finite volume and finite element methods for the shallow water equations with bottom topography and friction terms. Z Angew Math Mech 86:874
Lynett PJ, Wu TR, Liu PL (2002) Modeling wave runup with depth integrated equations. Coast Eng 46:89
Ma D-J, Sun D-J, Yin X-Y (2007) Hybrid finite element/volume method for shallow water equations. Int J Numer Methods Fluids 55:431
Marche F (2008) A simple well-balanced model for two-dimensional coastal engineering applications. In: Hyperbolic problems: theory, numerics, applications. Springer, Berlin/ Heidelberg, pp 271–283
Marche F, Bonneton P, Fabrie P, Seguin N (2007) Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes. Int J Numer Methods Fluids 53:867
Mavriplis DJ (2003) Revisiting the Least-squares procedure for gradient reconstruction on unstructured grids. NIA Report 2003-06, National Institute of Aerospace
Mavriplis DJ (2007) Unstructured mesh discritizations and solvers for computational aerodynamics. In: 18th AIAA computational fluid dynamics conference AIAA 2007-3955
Morton KW, Sonar T (2007) Finite volume methods for hyperbolic conservation laws. Acta Numer 16:155
Murillo J, Garcia-Navarro P, Burguete J (2009) Time step restrictions for well-balanced shallow water solutions in non-zero velocity steady states. Int J Numer Methods Fluids 60:1351
Murillo J, Garcia-Navarro P, Burguete J, Brufau P (2006) A conservative 2D model of inundation flow with solute transport over dry bed. Int J Numer Methods Fluids 52:1059
Murillo J, Garcia-Navarro P, Burguete J, Brufau P (2007) The influence of source terms on stability, accuracy and conservation in two-dimensional shallow flow simulation using triangular finite volumes. Int J Numer Methods Fluids 54:543
Namin M, Lin AA, Falconer B (2004) Modelling estuarine and coastal flows using an unstructured triangular finite volume algorithm. Adv Water Resour 27:1179
Nikolos IK, Delis AI (2009) An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography. Comput Methods Appl Mech Eng 198:3723
Noelle S, Pankratz N, Puppo G, Natvig JR (2006) Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J Comput Phys 213:474
Rhebergen S, Bokhove O, van der Vegt JJW (2008) Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. J Comput Phys 227:1887
Ribeiro FLB, Galeao AC, Landau L (2001) Edge-based finite element method for shallow water equations. Int J Numer Methods Fluids 36:659
Ricchiuto M, Abgrall R, Deconinck H (2007) Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes. J Comput Phys 222:287
Ricchiuto M, Bollermann A (2008) Accuracy of stabilized residual distribution for shallow water flows including dry beds. In: Proceedings of the 12th international conference on hyperbolic problems: theory, numerics, applications (HYP08), Maryland, USA, June 2008
Ricchiuto M, Bollermann A (2009) Stabilized residual distribution for shallow water simulations. J Comput Phys 1071:1115
Roe PL (1981) Aprroximate Riemann solvers, parameter vectors, and difference schemes. J Comput Phys 43:357
Roe PL (1992) Sonic flux formulae. SIAM J Sci Stat Comput 13:611
Rogers BD, Borthwick AGL, Taylor PH (2003) Mathematical balancing of flux gradient and source terms prior to using Roe’s approximate Riemann solver. J Comput Phys 192:422
Selmin V (1993) The node-centered finite volume approach: bridge between finite differences and finite elements. Comput Methods Appl Mech Eng 102:107
Serrano-Pacheco A, Murrillo J, Garcia-Navarro P (2009) Finite volume method for the simulation of the waves generated by landslides. J Hydrol 373:273
Sivakumar P, Hyams DG, Taylor LK, Briley WR (2009) A primitive-variable Riemann method for solution of the shallow water equations with wetting and drying. J Comput Phys 228:7452
Sun M, Takayama K (2003) Error localization in solution-adaptive grid methods. J Comput Phys 190:346
Sweby PK (1984) High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J Numer Anal 21:995
Thacker WC (1981) Some exact solutions to the nonlinear shallow water wave equations. J Fluid Mech 107:499
Thomas JL, Diskin B, Rumsey CL (2008) Towards verification of unstructured-grid solvers. AIAA paper 2008-666, 46th AIAA Aerospace sciences meeting and exhibit: Reno, Nevada
Toro EF (1997) Riemann solvers and numerical methods for fluid dynamics. Springer, Berlin
Toro EF (2001) Shock-capturing methods for free surface shallow flows. Wiley, Chichester
Toro EF, Garcia-Navarro P (2007) Godunov-type methods for free-surface shallow flows: a review. J Hydraul Res 45:736
Toro EF, Titarev V (2006) Derivative Riemann solvers for systems of conservation laws and ADER methods. J Comput Phys 212:150
Valiani A, Begnudelli L (2006) Divergence form for the bed slope source term in shallow water equations. ASCE J Hydraul Eng 132(7):652
Valiani A, Caleffi V, Zanni A (2002) Case study: Malpasset dam-break simulation using a two-dimensional finite volume method. ASCE J Hydraul Eng 128:460
Van Albada GD, Van Leer B, Roberts WW (1982) A comparative study of computational methods in cosmic gas dynamics. Astron Astrophys 108:46–84
van Leer B (1979) Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method. J Comput Phys 32:101
van Leer B (2006) Upwind and high-resolution methods for compressible flow: from donor cell to residual-disrtibution schemes. Commun Comput Phys 1, 192–206
Venkatakrishan V (1993) On the accuracy of limiters and convergence to steady state solutions. AIAA paper 1993-0880
Vingoli G, Titarev VA, Toro EF (2008) ADER schemes for the shallow water equations in channel with irregular bottom elevation. J Comput Phys 227:2463
Wang J-W, Liu R-X (2005) Combined finite volume-finite element method for shallow water equations. Comput Fluids 34:1199
Xing J, Shu CW (2005) High-order finite difference WENO schemes with exact conservation property for the shallow water equations. J Comput Phys 208:206
Ying X, Wang SSY, Khan AA (2003) Numerical simulation of flood inundation due to dam and levee breach. In: Bizier P, DeBarry P (eds) World water and environmental resources congress 2003. ASCE, p 366
Zhou JG, Causon DM, Ingram DM, Mingham CJ (2002) Numerical solutions of the shallow water equations with discontinuous bed topography. Int J Numer Methods Fluids 38:769
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Delis, A.I., Nikolos, I.K. & Kazolea, M. Performance and Comparison of Cell-Centered and Node-Centered Unstructured Finite Volume Discretizations for Shallow Water Free Surface Flows. Arch Computat Methods Eng 18, 57–118 (2011). https://doi.org/10.1007/s11831-011-9057-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831-011-9057-6