×

zbMATH — the first resource for mathematics

The MOOD method for the non-conservative shallow-water system. (English) Zbl 1390.76415
Summary: We present an adaptation of the MOOD method, initially introduced in [the first author et al., J. Comput. Phys. 230, No. 10, 4028–4050 (2011; Zbl 1218.65091); S. Diot et al., Comput. Fluids 64, 43–63 (2012; Zbl 1365.76149)], for the two-dimensional shallow-water system with varying bathymetry, where the major novelty of the study is the non-conservative term discretization in the framework of the MOOD strategy. We derive a robust sixth-order well-balanced scheme and propose a large panel of numerical tests to assess the accuracy of the method and show that numerical solutions are free of oscillations in the vicinity of discontinuities. We also demonstrate that the MOOD method guarantees the height positivity as long as the first-order scheme does.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
Software:
HLLE; Gmsh; SWASHES
PDF BibTeX Cite
Full Text: DOI
References:
[1] Clain, S.; Diot, S.; Loubère, R., A high-order finite volume method for hyperbolic systems: multi-dimensional optimal order detection (MOOD), J Comput Phys, 230, 10, 4028-4050, (2011) · Zbl 1218.65091
[2] Diot, S.; Clain, S.; Loubère, R., Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials, Comput Fluids, 64, 43-63, (2012) · Zbl 1365.76149
[3] Castro, C. E.; Toro, E. F.; käser, M., ADER scheme on unstructured meshes for shallow water: simulation of tsunami waves, Geophys J Int, 189, 1505-1520, (2012)
[4] Omira, R.; Baptista, M. A.; Miranda, J. M., Evaluating tsunami impact on the gulf of cadiz coast (northeast atlantic), Pure Appl Geophys, 168, 1033-1043, (2011)
[5] Wijetunge, J. J., Numerical simulation of the 2004 indian Ocean tsunami: case study of effect of sand dunes on the spatial distribution of inundation in hambantota, sri lanka, J Appl Fluid Mech, 3, 125-135, (2010)
[6] Brecht, D.; Noble, P., Mathematical justification of the shallow water model, Methods Appl Anal, 14, 87-118, (2007)
[7] Xing, Y., Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium, J Comput Phys, 257, 536-553, (2014) · Zbl 1349.76289
[8] Xing, Y.; Shu, C. W., High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, J Comput Phys, 214, 567-598, (2006) · Zbl 1089.65091
[9] Godunov, S. K., A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations, Math Sbornik, 47, 271-306, (1959) · Zbl 0171.46204
[10] Berthon, C.; Fouchet, F., Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, J Comput Phys, 231, 4993-5015, (2012) · Zbl 1351.76095
[11] Gallouët, T.; Hérard, J. M.; Seguin, N., Some approximate Godunov scheme to compute shallow-water equations with topography, Comput Fluids, 32, 479-513, (2003) · Zbl 1084.76540
[12] Nikolos, I. K.; Delis, A. I., An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography, Comput Methods Appl Mech Engrg, 198, 3723-3750, (2009) · Zbl 1230.76035
[13] Vukovic, S.; Sopta, L., ENO and WNO schemes with the exact conservation property for one-dimensional shallow water equations, J Comput Phys, 179, 593-621, (2002) · Zbl 1130.76389
[14] Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, J. R., Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J Comput Phys, 213, 474-499, (2006) · Zbl 1088.76037
[15] Noelle, S.; Xing, Y.; Shu, C.-W., High-order well-balanced finite volume WENO schemes for shallow water equations with moving water, J Comput Phys, 226, 29-58, (2007) · Zbl 1120.76046
[16] Gottlieb, S.; Shu, C. W., Total variation diminishing Runge-Kutta schemes, Math Compt, 67, 73-85, (1998) · Zbl 0897.65058
[17] Titarev, V. A.; Toro, E. F., ADER schemes for three-dimensional non-linear hyperbolic systems, J Comput Phys, 204, 715-736, (2005) · Zbl 1060.65641
[18] Dumbser, M.; Castro, M.; Parés, C.; Toro, E. F., ADER schemes on unstructured meshes for nonconservative hyperbolic systems: applications to geophysical flows, Comput Fluids, 38, 1731-1748, (2009) · Zbl 1177.76222
[19] Diot, S.; Loubère, R.; Clain, S., The MOOD method in the three-dimensional case: very-high-order finite volume method for hyperbolic systems, Int J Numer Meth Fluids, 73, 362-392, (2013)
[20] Berthon, C.; Desveaux, V., An entropy preserving MOOD scheme for the Euler equations, Int J finite volumes, 11, 1-39, (2014)
[21] Dumber, M.; Zanotti, O.; Loubère, R.; Diot, S., A posteriori subcell limiting for discontinuous Galerkin finite element method for hyperbolic system of conservation laws, J Comput Phys, 278, 47-75, (2014) · Zbl 1349.65448
[22] Loubère, R.; Dumbser, M.; Diot, S., A new family of high order unstructured MOOD and ADER finite volume schemes for multidimensional systems of hyperbolic conservation laws, Commun Comput Phys, 16, 718-763, (2014) · Zbl 1373.76137
[23] Diot, S.; François, M. M.; Dendy, E. D., A higher-order unsplit 2d direct Eulerian finite volume method for two-material compressible flows based on the MOOD paradigms, Int J Numer Meth Fluids, 3966, 2014, (2014)
[24] Buffard, T.; Clain, S., Monoslope and multislope MUSCL methods for unstructured meshes, J Comput Phys, 229, 3745-3776, (2010) · Zbl 1189.65204
[25] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J Comput Phys, 114, 45-58, (1994) · Zbl 0822.65062
[26] Maso, G. D.; LeFloch, P.; Murat, F., Definition and weak stability of nonconservative products, J Math Pures Appl, 74, 483-548, (1995) · Zbl 0853.35068
[27] Gosse, L., A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms, Comput Math Appl, 39, 135-159, (2000) · Zbl 0963.65090
[28] Castro, M.; Gallardo, J.; Parés, C., High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems, Math Comput, 75, 1103-1134, (2006) · Zbl 1096.65082
[29] Castro, M. J.; LeFloch, P. G.; noz Ruiz, M. L.M.; Parés, C., Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J Comput Phys, 227, 8107-8129, (2008) · Zbl 1176.76084
[30] Bermúdez, A.; Vázquez, M. E., Upwind methods for hyperbolic conservation laws with source terms, Comput Fluids, 24, 1049-1071, (1994) · Zbl 0816.76052
[31] Bermúdez, A.; Dervieux, A.; Desideri, J.-A.; Vásquez, M. E., Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput Methods Appl Mech Engrg, 155, 49-72, (1998) · Zbl 0961.76047
[32] LeVeque, R. J., Balancing source terms and flux gradients on high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J Comput Phys, 146, 346-365, (1998) · Zbl 0931.76059
[33] Hubbard, M.; Garcia-Navarro, P., Flux difference splitting and the balancing of source terms and flux gradients, J Comput Phys, 165, 89-125, (2000) · Zbl 0972.65056
[34] 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, 2050-2065, (2004) · Zbl 1133.65308
[35] Berthon, C.; Marche, F.; Turpault, R., An efficient scheme on wet/dry transitions for shallow water equations with friction, Comput Fluids, 48, 192-201, (2011) · Zbl 1271.76178
[36] Canestrelli, A.; Siviglia, A.; Dumbser, M.; Toro, E. F.-., Well-balanced high-order centered schemes for non-conservative hyperbolic systems. application to shallow water equations with fixed and mobile bed, Adv Water Resour, 32, 834-844, (2009)
[37] Caleffi, V.; Valiani, A.; Bernini, A., Fourth-order balanced sourced term treatment in central WENO schemes for shallow water equations, J Comput Phys, 218, 228-245, (2006) · Zbl 1158.76376
[38] Kurganov, A.; Levy, D., Central-upwind schemes for the Saint-Venant system, ESAIM: Math Model Num Anal, 36, 397-425, (2002) · Zbl 1137.65398
[39] Kurganov, A.; Petrova, G., A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system, Commun Math Sci, 5, 133-160, (2007) · Zbl 1226.76008
[40] Bryson, S.; Epshteyn, Y.; Kurganov, A.; Petrova, G., Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system, M2AN, Math Model Numer Anal, 45, 423-446, (2011) · Zbl 1267.76068
[41] Xing, Y.; Shu, C. W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J Comput Phys, 208, 206-227, (2005) · Zbl 1114.76340
[42] Castro Díaz, M. J.; López-García, J. A.; Parés, C., High order exactly well-balanced numerical methods for shallow water systems, J Comput Phys, 246, 242-264, (2013) · Zbl 1349.76315
[43] Xing, Y.; Shu, C. W.; Noelle, S., On the advantage of well-balanced schemes for moving water equilibria of the shallow water equations, J Sci Comput, 48, 339-349, (2011) · Zbl 1409.76086
[44] Alcrudo, F.; Benkhaldoum, F., Exact solution to the Riemann problem of the shallow water equations with bottom step, Comput Fluids, 30, 643-671, (2001) · Zbl 1048.76008
[45] Noussair, A., Riemann problem with nonlinear resonance effects and well-balanced Godunov scheme for shallow water fluid flow past an obstacle,, SIAM J Numer Anal, 39, 52-72, (2001) · Zbl 1001.35085
[46] Bernetti, R.; Titarev, V. A.; Toro, E. F., Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry, J Comput Phys, 22, 3212-3243, (2008) · Zbl 1132.76027
[47] LeFloch, P. G.; Thanh, M. D., A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime, J Comput Phys, 230, 7631-7660, (2011) · Zbl 1453.35150
[48] Thanh, M. D., Numerical treatment in resonant regime for shallow water equations with discontinuous topography, Commun Nonlinear Sci Numer Simulat, 18, 417-433, (2013) · Zbl 1322.76019
[49] Han, E.; Warnecke, G., Exact Riemann solutions to shallow water equations, Q Appl Math, 3, 72, 407-453, (2014) · Zbl 1298.76047
[50] Andrianov, N., Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations, Int J Numer Meth Fluids, 47, 825-831, (2005) · Zbl 1134.76402
[51] Zhou, J. G.; Causon, D. M.; Mingham, C. G.; Ingram, D. M., The surface gradient method for the treatment of source terms in the shallow-water equations, J Comput Phys, 168, 1-25, (2001) · Zbl 1074.86500
[52] Zhou, J. G.; Causon, D. M.; Mingham, C. G.; Ingram, D. M., Numerical solutions of the shallow water equations with discontinuous bed topography, Int J Numer Meth Fluids, 38, 769-788, (2002) · Zbl 1040.76045
[53] Duran, A.; Liang, Q.; Marche, F., On the well-balanced numerical discretization of shallow water equations on unstructured meshes, J Comput Phys, 235, 565-586, (2013) · Zbl 1291.76215
[54] Clain, S.; Machado, G.; Nóbrega, J. M.; Pereira, R., A sixth-order finite volume method for multidomain convection-diffusion problem with discontinuous coefficients, Comput Methods Appl Mech Eng, 267, 43-64, (2013) · Zbl 1286.80003
[55] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, 3rd revision, (2009), Springer-Verlag Berlin and Heidelberg GmbH & Co. K
[56] Harten, A.; Lax, P. D.; Leer, B. V., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev, 25, 35-61, (1983) · Zbl 0565.65051
[57] Toro, E. F.; Spruce, M.; Spares, W., Restoration of the contact surface in the HLL Riemann solver, Shock Wave, 4, 25-34, (1994) · Zbl 0811.76053
[58] Huang, Y.; Zhang, N.; Pei, Y., Well-balanced finite volume scheme for shallow water flooding and drying over arbitrary topography, Eng Appl Comput Fluid Mech, 7, 40-54, (2013)
[59] Geuzaine, C.; Remacle, J.-F., Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Int J Numer Meth Eng, 79, 1309-1331, (2009) · Zbl 1176.74181
[60] Berthon, C.; Chalons, C., A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. to appear in, Math Comp, 85, 1281-1307, (2016) · Zbl 1382.76180
[61] Delestre, O.; Lucas, C.; Ksinant, P.-A.; Darboux, F.; Laguerre, C.; Vo, T.-N.-T., SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies, Int J Numer Meth Fluids, 74, 229-230, (2014)
[62] Murillo, J.; a Navarro, P. G., Weak solutions for partial differential equations with source terms: application to the shallow water equations, J Comput Phys, 229, 4327-4368, (2010) · Zbl 1334.35014
[63] Murillo, J.; a Navarro, P. G., Augmented versions of the HLL and HLLC Riemann solvers including source terms in one and two dimensions for shallow water flow applications, J Comput Phys, 231, 6861-6906, (2012) · Zbl 1284.35118
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.