×

Fourth-order balanced source term treatment in central WENO schemes for shallow water equations. (English) Zbl 1158.76376

Summary: The aim of this work is to develop a well-balanced central weighted essentially non-oscillatory (CWENO) method, fourth-order accurate in space and time, for shallow water system of balance laws with bed slope source term. Time accuracy is obtained applying a Runge–Kutta scheme (RK), coupled with the natural continuous extension (NCE) approach. Space accuracy is obtained using WENO reconstructions of the conservative variables and of the water-surface elevation. Extension of the applicability of the standard CWENO scheme to very irregular bottoms, preserving high-order accuracy, is obtained introducing two original procedures. The former involves the evaluation of the point-values of the flux derivative, coupled with the bed slope source term. The latter involves the spatial integration of the source term, analytically manipulated to take advantage from the regularity of the free-surface elevation, usually smoother than the bottom elevation. Both these procedures satisfy the C-property, the property of exactly preserving the quiescent flow. Several standard one-dimensional test cases are used to verify high-order accuracy, exact C-property, and good resolution properties for smooth and discontinuous solutions.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
65D05 Numerical interpolation
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35A35 Theoretical approximation in context of PDEs
35Q35 PDEs in connection with fluid mechanics

Software:

CWENO
PDFBibTeX XMLCite
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, International Journal of Numerical Methods in Fluids, 16, 489-505 (1993) · Zbl 0766.76067
[2] Arminjon, P.; St-Cyr, A., Nessyahu Tadmor-type central finite volume methods without predictor for 3D cartesian and unstructured tetrahedral grids, Applied Numerical Mathematics, 46, 135-155 (2003) · Zbl 1025.65048
[3] Bermudez, A.; Vázquez-Cendón, M. E., Upwind methods for hyperbolic conservation laws with source terms, Computers and Fluids, 23, 8, 1049-1071 (1994) · Zbl 0816.76052
[4] Bryson, S.; Levy, D., Balanced central schemes for shallow water equations on unstructured grids, SIAM Journal of Scientific Computation, 27, 2, 532-552 (2005) · Zbl 1089.76036
[5] Caleffi, V.; Valiani, A.; Zanni, A., Finite volume method for simulating extreme flood events in natural channels, IAHR Journal of Hydraulic Research, 41, 2, 167-177 (2003)
[6] Fraccarollo, L.; Capart, H., Riemann wave description of erosional dam-break flows, Journal of Fluid Mechanics, 461, 183-228 (2002) · Zbl 1142.76344
[7] Garcia-Navarro, P.; Vázquez-Cendón, M. E., On numerical treatment of the source terms in shallow water equations, Computers and Fluids, 29, 951-979 (2000) · Zbl 0986.76051
[8] Godunov, S. K., A difference scheme for numerical computation of discontinuous solution of hydrodynamic equations, Mathematics of the USSR-Sbornik, 43, 271-306 (1959) · Zbl 0171.46204
[9] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S. R., Uniformly high-order accurate essentially non-oscillatory schemes, III, Journal of Computational Physics, 71, 2, 231-475 (1987) · Zbl 0652.65067
[10] Hubbard, M. E.; Garcia-Navarro, P., Flux difference splitting and the balancing of source terms and flux gradients, Journal of Computational Physics, 165, 1, 89-125 (2000) · Zbl 0972.65056
[11] Jansen, P. P.; Van Bendegom, L.; Van den Berg, J.; De Vries, M.; Zanen, A., Principles of River Engineering, the Non-Tidal Alluvial River (1979), Pitman Publishing Ltd.: Pitman Publishing Ltd. London
[12] Jiang, G. S.; Levy, D.; Lin, C. T.; Osher, S.; Tadmor, E., High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws, SIAM Journal on Numerical Analysis, 35, 6, 2147-2168 (1998) · Zbl 0920.65053
[13] Jiang, G. S.; Shu, C. W., Efficient implementation of weighted ENO schemes, Journal of Computational Physics, 126, 1, 202-228 (1996) · Zbl 0877.65065
[14] Jiang, G. S.; Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM Journal on Scientific Computing, 19, 6, 1892-1917 (1998) · Zbl 0914.65095
[15] Kikkawa, H.; Ikeda, S.; Kitagawa, A., Flow and bed topography in curved open channels, ASCE Journal of Hydraulics Division, 102, 9, 1327-1342 (1976)
[16] Kurganov, A.; Noelle, S.; Petrova, G., Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM Journal of Scientific Computation, 23, 707-740 (2001) · Zbl 0998.65091
[17] Kurganov, A.; Petrova, G., Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws, Numerical Methods for Partial Differential Equations, 21, 3, 536-552 (2005) · Zbl 1071.65122
[18] Lax, P. D., Weak solutions of nonlinear hyperbolic equations and their numerical computations, Communications of Pure and Applied Mathematics, 44, 21-41 (1954)
[19] LeVeque, R. J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, Journal of Computational Physics, 146, 1, 346-365 (1998) · Zbl 0931.76059
[20] Levy, D.; Puppo, G.; Russo, G., Central WENO schemes for hyperbolic systems of conservation laws, Mathematical Modelling and Numerical Analysis, 33, 3, 547-571 (1999) · Zbl 0938.65110
[21] Levy, D.; Puppo, G.; Russo, G., A fourth-order central WENO scheme for multidimensional hyperbolic systems of conservation laws, SIAM Journal on Scientific Computing, 24, 2, 480-506 (2002) · Zbl 1014.65079
[22] Liggett, J. A., Fluid Mechanics (1994), McGraw-Hill: McGraw-Hill New York
[23] Liu, X. D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115, 1, 200-212 (1994) · Zbl 0811.65076
[24] Mingham, C. G.; Causon, D. M., High-resolution finite-volume method for shallow water flows, ASCE Journal of Hydraulic Engineering, 124, 6, 605-614 (1998)
[25] Nessyahu, H.; Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws, Journal of Computational Physics, 87, 2, 408-463 (1990) · Zbl 0697.65068
[26] S. Noelle, W. Rosenbaum, M. Rumpf, 3D adaptive central schemes: Part I. Algorithms for assembling the dual mesh, Applied Numerical Mathematics (accepted for publication).; S. Noelle, W. Rosenbaum, M. Rumpf, 3D adaptive central schemes: Part I. Algorithms for assembling the dual mesh, Applied Numerical Mathematics (accepted for publication). · Zbl 1137.76041
[27] Pareschi, L.; Puppo, G.; Russo, G., Central Runge-Kutta schemes for conservation laws, SIAM Journal on Scientific Computing, 26, 3, 979-999 (2005) · Zbl 1077.65094
[28] Qiu, J.; Shu, C. W., On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes, Journal of Computational Physics, 183, 1, 187-209 (2002) · Zbl 1018.65106
[29] I.L. Rozovskii, Flow of water in bends of open channels. Technical Report, Academy of Sciences of the Ukrainian S.S.R., Institute of Hydrology and Hydraulic Engineering, Kiev, 1957.; I.L. Rozovskii, Flow of water in bends of open channels. Technical Report, Academy of Sciences of the Ukrainian S.S.R., Institute of Hydrology and Hydraulic Engineering, Kiev, 1957.
[30] C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE Technical Report 97-65, NASA, Langley Research Center, Hampton, VA, USA, 1997.; C.W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, ICASE Technical Report 97-65, NASA, Langley Research Center, Hampton, VA, USA, 1997. · Zbl 0927.65111
[31] Soulis, J. V., Computation of two-dimensional dambreak flood flows, International Journal of Numerical Methods in Fluids, 14, 631-664 (1992) · Zbl 0825.76078
[32] Toro, E., Riemann problems and the WAF method for solving the two-dimensional shallow water equations, Philosophical Transactions of the Royal Society, A338, 43-68 (1992) · Zbl 0747.76027
[33] Črnjarić-Žic, N.; Vuković, S.; Sopta, L., Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equations, Journal of Computational Physics, 200, 2, 512-548 (2004) · Zbl 1115.76364
[34] A. Valiani, L. Begnudelli, Divergence form for bed slope source term (DFB) in shallow water equations, ASCE Journal of Hydraulic Engineering (2006) (to appear).; A. Valiani, L. Begnudelli, Divergence form for bed slope source term (DFB) in shallow water equations, ASCE Journal of Hydraulic Engineering (2006) (to appear).
[35] A. Valiani, V. Caleffi, A. Bernini, Central WENO schemes for shallow water movable bed equations, in: 12th International Conference on Transport and Sedimentation of Solid Particles, vol. 2, Prague, Czech Republic, September 2004, Institute of Hydrodynamics, Academy of Sciences of the Czech Republic, pp. 651-658.; A. Valiani, V. Caleffi, A. Bernini, Central WENO schemes for shallow water movable bed equations, in: 12th International Conference on Transport and Sedimentation of Solid Particles, vol. 2, Prague, Czech Republic, September 2004, Institute of Hydrodynamics, Academy of Sciences of the Czech Republic, pp. 651-658. · Zbl 1158.76376
[36] Valiani, A.; Caleffi, V.; Zanni, A., Case study: Malpasset dambreak simulation using a 2D finite volume method, ASCE Journal of Hydraulic Engineering, 128, 5, 460-472 (2002)
[37] Vázquez-Cendón, M. E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, Journal of Computational Physics, 148, 2, 497-526 (1999) · Zbl 0931.76055
[38] Vuković, S.; Sopta, L., ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations, Journal of Computational Physics, 179, 2, 593-621 (2002) · Zbl 1130.76389
[39] Vuković, S.; Črnjarić-Žic, N.; Sopta, L., WENO schemes for balance laws with spatially varying flux, Journal of Computational Physics, 199, 1, 87-109 (2004) · Zbl 1127.76344
[40] Xing, Y.; Shu, C. W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, Journal of Computational Physics, 208, 1, 206-227 (2005) · Zbl 1114.76340
[41] Zennaro, M., Natural continuous extension of Runge-Kutta methods, Mathematics of Computation, 46, 119-133 (1986) · Zbl 0608.65043
[42] Zhou, J. G.; Causon, D. M.; Ingram, D. M.; Mingham, C. G., The surface gradient method for the treatment of source terms in the shallow-water equations, Journal of Computational Physics, 168, 1, 1-25 (2001) · Zbl 1074.86500
[43] Zhou, J. G.; Causon, D. M.; Ingram, D. M.; Mingham, C. G., Numerical solutions of the shallow water equations with discontinuous bed topography, International Journal for Numerical Methods in Fluids, 38, 8, 769-788 (2002) · Zbl 1040.76045
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.