A new central-upwind scheme for solving the shallow water equations

Document Type : Original Article

Author

School of Engineering Science, College of Engineering, University of Tehran, Iran

Abstract

We have conducted a study where we applied one-dimensional central-upwind methods to estimate solutions of the Saint-Venant (SV) system. Our approach carefully considers the source terms associated with bottom topography. Within the context of the SV system, there are steady-state solutions that arise when the non-zero gradients of flux are exactly balanced by the corresponding source terms. Maintaining this delicate equilibrium with numerical approaches
presents a challenging problem. Finding slight variations in these states proves to be extremely difficult in the field of computing. To address this, we propose an extension of semi-discrete central schemes, commonly employed in hyperbolic conservation law systems, to encompass balance laws (BL). In our approach, we specifically focus on discretizing the source term with great care and attention. To verify the superior accuracy, precise preservation of the C-property, and outstanding resolution of our approach, we perform comprehensive one-dimensional simulations on both continuous and discontinuous solutions. 

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References

[1] R. Abedian, High-order semi-discrete central-upwind schemes with Lax-Wendroff-type time discretizations for Hamilton-Jacobi equations, Comput. Methods Appl. Math., 18 (2018), pp. 559–580.
[2] , A new high-order weighted essentially non-oscillatory scheme for non-linear degenerate parabolic equa[1]tions, Numer. Methods Partial Differential Equations, 37 (2021), pp. 1317–1343.
[3] , A finite difference Hermite RBF-WENO scheme for hyperbolic conservation laws, Internat. J. Numer.Methods Fluids, 94 (2022), pp. 583–607.
[4] , A modified high-order symmetrical WENO scheme for hyperbolic conservation laws, Int. J. Nonlinear Sci. Numer. Simul., 24 (2023), pp. 1521–1538.
[5] R. Abedian and M. Dehghan, The formulation of finite difference RBFWENO schemes for hyperbolic conservation laws: an alternative technique, Adv. Appl. Math. Mech., 15 (2023), pp. 1023–1055.
[6] E. Audusse, M.-O. Bristeau, and B. Perthame, Kinetic schemes for Saint-Venant equations with source terms on unstructured grids, Research Report RR-3989, INRIA, 2000. Projet M3N.
[7] A. Bermudez and M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. & Fluids, 23 (1994), pp. 1049–1071.
[8] K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. U.S.A., 68 (1971), pp. 1686–1688.
[9] T. Gallouet, J.-M. H ¨ erard, and N. Seguin ´ , Some approximate Godunov schemes to compute shallowwater equations with topography, Comput. & Fluids, 32 (2003), pp. 479–513.
[10] A. Kurganov and D. Levy, A third-order semidiscrete central scheme for conservation laws and convection[1]diffusion equations, SIAM J. Sci. Comput., 22 (2000), pp. 1461–1488.
[11] , Central-upwind schemes for the Saint-Venant system, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 397–425.
[12] A. Kurganov, S. Noelle, and G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and hamilton–jacobi equations, SIAM J. Sci. Comput., 23 (2001), pp. 707–740.
[13] A. Kurganov and G. Petrova, Central schemes and contact discontinuities, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 1259–1275.
[14] , A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., 88 (2001), pp. 683–729.
[15] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160 (2000), pp. 241–282.
[16] R. J. LeVeque, Balancing source terms and flux gradients in high-resolution Godunov methods: the quasisteady wave-propagation algorithm, J. Comput. Phys., 146 (1998), pp. 346–365.
[17] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), pp. 408–463.
[18] S. Rathan, N. R. Gande, and A. A. Bhise, Simple smoothness indicator WENO-Z scheme for hyperbolic conservation laws, Appl. Numer. Math., 157 (2020), pp. 255–275.
[19] S. Rathan and G. Naga Raju, A modified fifth-order WENO scheme for hyperbolic conservation laws, Comput. Math. Appl., 75 (2018), pp. 1531–1549.
[20] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), pp. 439–471.
[21] S. Vukovic and L. Sopta, ENO and WENO schemes with the exact conservation property for one dimensional shallow water equations, J. Comput. Phys., 179 (2002), pp. 593–621.
[22] Z. Zhao, Y. Chen, and J. Qiu, A hybrid Hermite WENO scheme for hyperbolic conservation laws, J. Comput. Phys., 405 (2020), pp. 109175, 22.
[23] Z. Zhao and J. Qiu, A Hermite WENO scheme with artificial linear weights for hyperbolic conservation laws, J. Comput. Phys., 417 (2020), pp. 109583, 23.20
AUT Journal of Mathematics and Computing
Volume 6, Issue 3
2025
Pages 193-203

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