×

zbMATH — the first resource for mathematics

Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods. (English) Zbl 1067.76586
Summary: A generalization and extension of a finite difference method for calculating numerical solutions of the two dimensional shallow water system of equations is investigated. A previously developed non-oscillatory relaxation scheme is generalized as to included problems with source terms in two dimensions, with emphasis given to the bed topography, resulting in a class of methods of first- and second-order in space and time. The methods are based on classical relaxation models combined with TVD Runge-Kutta time stepping mechanisms where neither Riemann solvers nor characteristic decompositions are needed. Numerical results are presented for several test problems with or without the source term present. The wetting and drying process is also considered. The presented schemes are verified by comparing the results with documented ones.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alcrudo, F.; Garcia-Navarro, P., A high resolution Godunov-type scheme in finite volumes for the 2D shallow water equations, Int. J. numer. methods fluids, 16, 489-505, (1993) · Zbl 0766.76067
[2] Aregba-Driollet, D.; Natalini, R., Convergence of relaxation schemes for conservation laws, Appl. anal., 61, 163-193, (1996) · Zbl 0887.65100
[3] Arvanitis, Ch.; Katsaounis, Th.; Makridakis, Ch., Adaptive finite element relaxation schemes for hyperbolic conservations laws, M2an, 35, 17-33, (2001) · Zbl 0980.65104
[4] Bermudez, A.; Dervieux, A.; Desideri, J.A.; Vazquez, M.E., Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. methods appl. mech. eng., 155, 49-72, (1998) · Zbl 0961.76047
[5] Brufau, P.; Garcia-Navarro, P., Unsteady free surface flow simulation over complex topography with a multidimensional upwind technique, J. comp. phys., 186, 503-526, (2003) · Zbl 1047.76537
[6] Brufau, P.; Vázquez-Céndon, M.E.; Garcia-Navarro, P., A numerical model for the flooding and drying of irregular domains, Int. J. numer. methods fluids, 39, 247-275, (2002) · Zbl 1094.76538
[7] Chalabi, A., Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms, Math. comput., 68, 955-970, (1999) · Zbl 0918.35088
[8] Chalabi, A.; Seghir, D., Convergence of relaxation schemes for initial boundary value problems for conservation laws, Comp. math. appl., 43, 1079-1093, (2002) · Zbl 1050.65081
[9] Chen, G.O.; Liu, T.P., Zero relaxation and dissipation limits for hyperbolic conservation laws, Commun. pure appl. math., 46, 744-781, (1993) · Zbl 0797.35113
[10] Chen, G.O.; Levermore, C.D.; Liu, T.P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. pure appl. math., 47, 787-830, (1994) · Zbl 0806.35112
[11] Delis, A.I., Improved application of the HLLE Riemann solver for the shallow water equations with source terms, Commun. numer. methods eng., 39, 59-83, (2003) · Zbl 1058.76041
[12] Delis, A.I.; Skeels, C.P., TVD schemes for open channel flow, Int. J. numer. methods fluids, 26, 791-809, (1998) · Zbl 0927.76067
[13] Delis, A.I.; Skeels, C.P.; Ryrie, S.C., Implicit high-resolution methods for modelling one-dimensional open channel flow, J. hydraul. res., 40, 369-382, (2000)
[14] Delis, A.I.; Katsaounis, Th., Relaxation schemes for the shallow water equations, Int. J. numer. methods fluids, 41, 695-719, (2003) · Zbl 1023.76031
[15] Fennema, R.J.; Chaudhry, M.H., Explicit methods for 2D transient free surface flows, J. hydraul. eng.-ASCE, 116, 1013-1034, (1990)
[16] Garcia-Navarro, P.; Vázquez-Cendón, M.E., On numerical treatment of the source terms in the shallow water equations, Comput. fluids, 29, 951-979, (2000) · Zbl 0986.76051
[17] Jin, S.; Xin, Z., The relaxing schemes of conservations laws in arbitrary space dimensions, Commun. pure appl. math., 48, 235-277, (1995)
[18] Jin, S.; Levermore, C.D., Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. comp. phys, 126, 449-467, (1996) · Zbl 0860.65089
[19] Katsaounis, Th.; Makridakis, Ch., Finite volume relaxation schemes for the multidimensional conservation laws, Math. computat., 70, 533-553, (2001) · Zbl 0967.65094
[20] Katsaounis, Th.; Makridakis, Ch., Adaptive finite element relaxation schemes for the Saint-Venant system, (), 621-631 · Zbl 1134.76387
[21] Katsoulakis, M.; Kossioris, G.; Makridakis, Ch., Convergence and error estimates of relaxation schemes for multidimensional conservation laws, Commun. part. different. equat., 24, 395-424, (1999) · Zbl 0926.35088
[22] Lattanzio, C.; Serre, D., Convergence of a relaxation scheme for hyperbolic systems of conservation laws, Numer. math., 88, 121-134, (2001) · Zbl 0983.35086
[23] LeVêque, R.J.; Pelanti, M., A class of approximated Riemann solvers and their relation to relaxation schemes, J. comput. phys., 172, 572-591, (2001) · Zbl 0988.65072
[24] LeVêque, R.J., Finite volume methods for hyperbolic problems, (2002), Cambridge University Press Cambridge · Zbl 1010.65040
[25] Liu, T.P., Hyperbolic conservation laws with relaxation, Commun. math. phys., 108, 153-175, (1987) · Zbl 0633.35049
[26] Liu, H.L.; Warnecke, G., Convergence rates for relaxation schemes approximating conservation laws, SIAM J. numer. anal., 37, 1316-1337, (2000) · Zbl 0954.35108
[27] Louaked, M.; Hanich, L., TVD-multiresolution scheme for the shallow water equations, C.R. acad. sci. Paris sr. I math., 331, 745-750, (2000) · Zbl 1010.76069
[28] Louaked, M.; Hanich, L., TVD scheme for the shallow water equations, J. hydraulic res., 363-378, (1998)
[29] Mingham, C.G.; Causon, D.M., High-resolution finite-volume method for shallow water flows, J. hydraul. eng.-ASCE, 124, 605-614, (1998)
[30] Natalini, R., Convergence to equilibrium for the relaxation approximations of conservation laws, Commun. pure appl. math., 49, 795-823, (1996) · Zbl 0872.35064
[31] Rebollo, T.C.; Delgado, A.D.; Nieto, E.D.F., A family of stable numerical solvers for the shallow water equations with source terms, Comput. methods appl. mech. eng., 192, 203-225, (2003) · Zbl 1083.76557
[32] L. Pareschi, G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation, J. Sci. Comput., in press. · Zbl 1203.65111
[33] Schroll, H.J., High resolution relaxed upwind schemes in gas dynamics, J. scientific computing, 17, 599-607, (2002) · Zbl 1002.76086
[34] Shu, C.-W.; Osher, S., Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072
[35] Sweby, P.K., High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. numer. anal., 21, 995-1011, (1984) · Zbl 0565.65048
[36] Tang, T.; Wang, J., Convergence of MUSCL relaxing schemes to the relaxed schemes for conservation laws with stiff source terms, J. sci. comput., 15, 173-195, (2000) · Zbl 0982.65101
[37] Toro, E.F., Shock-capturing methods for free-surface shallow flows, (2001), Wiley New York · Zbl 0996.76003
[38] Tzavaras, A.E., Materials with internal variables and relaxation to conservation laws, Arch. ration. mech. anal., 146, 129-155, (1999) · Zbl 0973.74005
[39] Vázquez-Cendón, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. comp. phys., 148, 497-526, (1999) · Zbl 0931.76055
[40] Wang, J.S.; Ni, H.G.; He, Y.S., Finite-difference TVD scheme for computation of dam-break problems, J hydraul. eng.-ASCE, 126, 253-262, (2000)
[41] Wang, W.-C., Nonlinear stability of centered rarefaction waves of the jin-xin relaxation model for 2×2 conservation laws, Electron. J. different. equat., 57, 1-20, (2002) · Zbl 1007.65065
[42] Xu, W.Q., Relaxation limit for piecewise smooth solutions to systems for conservation laws, J. differen. equat., 162, 140-173, (2000) · Zbl 0949.35088
[43] Zoppou, C.; Roberts, S., Numerical solution of the two-dimensional unsteady dam break, Appl. math. model., 24, 457-475, (2000) · Zbl 1004.76064
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.