×

High order well-balanced finite difference WENO schemes for shallow water flows along channels with irregular geometry. (English) Zbl 1433.76120

Summary: In this paper, we present high order finite difference weighted essentially non-oscillatory (WENO) schemes for the shallow water flows along open channels with irregular geometry and over a non-flat bottom topography. The proposed schemes maintain the well-balanced property for the still water steady state solutions, namely preserve steady state at the discrete level, when there is a exact balance between the flux gradient and the source term. Compared with the traditional shallow water equations with constant cross section, the construction of the well-balanced schemes is not a trivial work due to the effect induced by the irregular geometry of the channels. To preserve the well-balanced property, we first reformulate the source term, then propose to construct the numerical fluxes by means of a flux modification technique, and finally discrete the source term with the help of a novel source term approximation. Benchmark numerical examples are applied to validate the good performances of the resulting schemes: well-balanced property, high order accuracy, and high resolution for the discontinuous solutions.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bradford, S. F.; Sanders, B. F., Finite volume model for shallow water flooding of arbitrary topography, J. Hydraul. Eng., 128, 3, 289-298 (2002)
[2] Gottardi, G.; Venutelli, M., Central scheme for the two-dimensional dam-break flow simulation, Adv. Water Resour., 27, 259-268 (2004)
[3] Vreugdenhil, C. B., Numerical Methods for Shallow-Water Flow, 15-25 (1995), Springer: Springer Dordrecht
[4] Castro, M. J.; García-Rodríguez, J. A.; González-Vida, J. M.; Macías, J.; Parés, C.; Vázquez-Cendón, M. E., Numerical simulation of two-layer shallow water flows through channels with irregular geometry, J. Comput. Phys., 195, 202-235 (2004) · Zbl 1087.76077
[5] Greenberg, J. M.; Leroux, A. Y., A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33, 1-16 (1996) · Zbl 0876.65064
[6] Greenberg, J. M.; Leroux, A. Y.; Baraille, R.; Noussair, A., Analysis and approximation of conservation laws with source terms, SIAM J. Numer. Anal., 34, 1980-2007 (1997) · Zbl 0888.65100
[7] Noelle, S.; Xing, Y. L.; Shu, C. W., High-order well-balanced schemes, Numerical Methods for Balance Laws (G. Puppo and G. Russo eds) (2010), Quaderni di Matematica
[8] Xing, Y. L.; 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
[9] 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
[10] Perthame, B.; Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term, Calcolo, 38, 201-231 (2001) · Zbl 1008.65066
[11] Xu, K., A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J. Comput. Phys., 178, 533-562 (2002) · Zbl 1017.76071
[12] 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
[13] 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
[14] 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
[15] Li, G.; Caleffi, V.; Gao, J. M., High-order well-balanced central WENO scheme for pre-balanced shallow water equations, Comput. Fluids, 99, 182-189 (2014) · Zbl 1391.76484
[16] Li, G.; Caleffi, V.; Qi, Z., A well-balanced finite difference WENO scheme for shallow water flow model, Appl. Math. Comput., 265, 1-16 (2015) · Zbl 1410.76296
[17] Luo, J.; Xu, K.; Liu, N., A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field, SIAM J. Sci. Comput., 33, 2356-2381 (2011) · Zbl 1232.76044
[18] Xing, Y.; Shu, C. W., High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields, J. Sci. Comput., 54, 645-662 (2013) · Zbl 1260.76022
[19] Käppeli, R.; Mishra, S., Well-balanced schemes for the euler equations with gravitation, J. Comput. Phys., 259, 199-219 (2014) · Zbl 1349.76345
[20] Chandrashekar, P.; Klingenberg, C., A second order well-balanced finite volume scheme for euler equations with gravity, SIAM J. Sci. Comput., 7, 3, B382-B402 (2015) · Zbl 1320.76078
[21] Li, G.; Xing, Y., Well-balanced discontinuous Galerkin methods for the euler equations under gravitational fields, J. Sci. Comput., 67, 493-513 (2016) · Zbl 1381.76184
[22] Li, G.; Xing, Y., High order finite volume WENO schemes for the euler equations under gravitational fields, J. Comput. Phys., 316, 145-163 (2016) · Zbl 1349.76356
[23] Ghosh, D.; Constantinescu, E. M., Well-balanced, conservative finite difference algorithm for atmospheric OWS, AIAA J., 54, 1370-1385 (2016)
[24] Li, G.; Xing, Y., Well-balanced discontinuous Galerkin methods with hydrostatic reconstruction for the euler equations with gravitation, J. Comput. Phys., 352, 445-462 (2018) · Zbl 1375.76089
[25] Vázquez-Céndon, M. E., Improved treatmeat of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys., 148, 497-526 (1999) · Zbl 0931.76055
[26] Bermudez, A.; Vazquez, M. E., Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids, 23, 1049-1071 (1994) · Zbl 0816.76052
[27] Garcia-Navarro, P.; Vázquez-Céndon, M. E., On numerical treatment of the source terms in the shallow water equations, Comput. Fluids, 29, 951-979 (2000) · Zbl 0986.76051
[28] Balbas, J.; Karni, S., A central scheme for shallow water flows along channels with irregular geometry, ESAIM: Math. Model. Numer. Anal., 43, 333-351 (2009) · Zbl 1159.76026
[29] Hernández-Duenas, G.; Karni, S., Shallow water flows in channels, J. Sci. Comput., 48, 190-208 (2011) · Zbl 1426.76377
[30] Balbas, J.; Hernandez-Duenas, G., A positivity preserving central scheme for shallow water flows in channels with wet-dry states, ESAIM: Math. Model. Numer. Anal., 48, 665-696 (2014) · Zbl 1433.76100
[31] Murillo, J.; García-Navarro, P., Accurate numerical modeling of 1D flow in channels with arbitrary shape. application of the energy balanced property, J. Comput. Phys., 260, 222-248 (2014) · Zbl 1349.76032
[32] Hernandez-Duenas, G.; Beljadid, A., A central-upwind scheme with artificial viscosity for shallow-water flows in channels, Adv. Water Resour., 96, 323-338 (2016)
[33] Xing, Y., High order finite volume WENO schemes for the shallow water flows through channels with irregular geometry, 299, 229-244 (2016) · Zbl 1382.76190
[34] Hill, D. J.; Pullin, D. I., Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks, J. Comput. Phys., 194, 435-450 (2004) · Zbl 1100.76030
[35] Pirozzoli, S., Conservative hybrid compact-WENO schemes for shock-turbulence interaction, J. Comput. Phys., 178, 81-117 (2002) · Zbl 1045.76029
[36] Taylor, E. M.; Wu, M.; Martín, M. P., Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulenc, J. Comput. Phys., 223, 384-397 (2007) · Zbl 1165.76350
[37] Colonius, T.; Lele, S. K., Computational aeroacoustics: progress on nonlinear problems of soundgeneration, Prog. Aerosp. Sci., 40, 345-416 (2004)
[38] C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, in: B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor, A. Quarteroni (Eds.), Lecture Notes in Math. 1697, Springer-Verlag, Berlin, 1998, pp. 325-432.; C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, in: B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor, A. Quarteroni (Eds.), Lecture Notes in Math. 1697, Springer-Verlag, Berlin, 1998, pp. 325-432. · Zbl 0927.65111
[39] Shu, C. W., High order weighted essentially nonoscillatory schemes for convection dominated problems, SIAM Rev., 51, 82-126 (2009) · Zbl 1160.65330
[40] Shu, C. W., High order WENO and DG methods for time-dependent convection-dominated PDEs: Abrief survey of several recent developments, J. Comput. Phys., 316, 598-613 (2016) · Zbl 1349.65486
[41] Jiang, G.; Shu, C. W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228 (1996) · Zbl 0877.65065
[42] Shu, C. W., Total-variation-diminishing time discretizations, SIAM J. Sci. Stat. Comput., 9, 1073-1084 (1988) · Zbl 0662.65081
[43] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43, 357-372 (1981) · Zbl 0474.65066
[44] 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
[45] Kurganov, A.; Levy, D., Central-upwind schemes for the Saint-Venant system, M2AN Math. Model. Numer. Anal., 36, 397-425 (2002) · Zbl 1137.65398
[46] García-Navarro, P.; Alcrudo, F.; Savirón, J. M., 1D open-channel flow simulation using TVD-mccormack scheme, J. Hydraul. Eng., 118, 1359-1372 (1992)
[47] Vazquez-Cendon, M. E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comput. Phys., 148, 497-526 (1999) · Zbl 0931.76055
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.