Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T01:05:19.618Z Has data issue: false hasContentIssue false
  • English
  • Français

High Order Well-Balanced Weighted Compact Nonlinear Schemes for Shallow Water Equations

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  28 July 2017

Zhen Gao  and
Guanghui Hu
Zhen Gao*
Affiliation:
School of Mathematical Sciences, Ocean University of China, Qingdao, China
Guanghui Hu*
Affiliation:
Department of Mathematics, University of Macau, Macao SAR, China UM Zhuhai Research Institute, Zhuhai, Guangdong, China
*
*Corresponding author. Email addresses:zhengao@ouc.edu.cn (Z. Gao), garyhu@umac.mo (G. H. Hu)
*Corresponding author. Email addresses:zhengao@ouc.edu.cn (Z. Gao), garyhu@umac.mo (G. H. Hu)
Get access

Abstract

In this study, a numerical framework of the high order well-balanced weighted compact nonlinear (WCN) schemes is proposed for the shallow water equations based on the work in [S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321]. We employ a special splitting technique for the source term proposed in [Y. Xing, C.-W Shu, J. Comput. Phys. 208 (2005) 206-227] to maintain the exact C-property, which can be proved theoretically. In the meantime, the genuine high order accuracy of the numerical scheme can be observed successfully, and small perturbation of the stationary state can be resolved and evolved well. In order to capture the strong discontinuities and large gradients, the fifth-order upwind weighted nonlinear interpolations together with the fourth/sixth order cell-centered compact scheme are used to construct different WCN schemes. In addition, the local characteristic projections are considered to further restrain the potential numerical oscillations. A variety of representative one- and two-dimensional examples are tested to demonstrate the good performance of the proposed schemes.

Keywords

Shallow water equationsC-propertyweighted compact nonlinear schemesource term

MSC classification

Secondary: 35L65: Conservation laws 35L67: Shocks and singularities
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 4 , October 2017 , pp. 1049 - 1068
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alcrudo, F. and Benkhaldoun, F., Exact solutions to the Riemann problem of the shallow water equations with a bottom step, Comput. Fluids. 30 (2001) 643–671.CrossRefGoogle Scholar
[2] Bermudez, A. and Vazquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids. 23 (1994) 1049–1071.Google Scholar
[3] Borges, R., Carmona, M., Costa, B. and Don, W.S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws, J. Comput. Phys. 227 (2008) 3101–3211.CrossRefGoogle Scholar
[4] Castro, M., Costa, B. and Don, W.S., High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws, J. Comput. Phys. 230 (2011) 1766–1792.Google Scholar
[5] Crnjaric-Zic, N., Vukovic, S. and Sopta, L., Extension of ENO and WENO schemes to one-dimensional sediment transport equations, Comput. Fluids 33 (2004) 31–56.CrossRefGoogle Scholar
[6] Deng, J., Li, R., Sun, T. and Wu, S., Robust a simulation for shallow flows with friction on rough topography, Numer. Math. Theor. Meth. Appl. 6 (2013) 384–407.CrossRefGoogle Scholar
[7] Deng, X., Mao, M., Jiang, Y. and liu, H., New high-order hybrid cell-edge and cell-node weighted compact nonlinear scheme. In: Proceedings of 20th AIAA CFD conference, AIAA 2011-3847, June 27-30, Honoluu, HI, USA, 2011.Google Scholar
[8] Deng, X. and Maekawa, H., Compact high-order accurate nonlinear schemes, J. Comput. Phys. 130 (1997) 77–91.Google Scholar
[9] Deng, X. and Zhang, H., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys. 165 (2000) 22–44.CrossRefGoogle Scholar
[10] Don, W.S., Gao, Z., Li, P. and Wen, X., Hybrid Compact-WENO Finite Difference Scheme with Conjugate Fourier Shock Detection Algorithm for Hyperbolic Conservation Laws, SIAM, J. Sci. Comput. 38(2) (2016) A691–A711.CrossRefGoogle Scholar
[11] Ern, A., Piperno, S. and Djadel, K., A well-balanced Runge-Kutta discontinuous Galerkin method for the shallow-water equations with flooding and drying, Int. J. Numer. Methods Fluids 58 (2008) 1–25.CrossRefGoogle Scholar
[12] Gandham, R. and Medina, D. and Warburton, T., GPU Accelerated Discontinuous Galerkin Methods for Shallow Water Equations, Commun. Comput. Phys. 18 (2015) 37–64.Google Scholar
[13] Iida, R., Asahara, M., Hayashi, A.K., Tsuboi, N. and Nonomura, T., Implementation of a Robust Weighted Compact Nonlinear Scheme for Modeling of Hydrogen/Air Detonation, Combut. Sci. Tech. 186 (2014) 1736–1757.Google Scholar
[14] Jiang, G.S. and Shu, C.-W., Efficient Implementation of Weighted ENO Schemes, J. Comput. Phys. 126 (1996) 202–228.Google Scholar
[15] Johnsen, E., Larsson, J., Bhagatwala, A.V., Cabot, W.H., Moin, P., Olson, B.J., et al. Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves, J. Comput. Phys. 229(4) (2010) 1213–37.CrossRefGoogle Scholar
[16] Kuang, D.Y. and Lee, L., A conservative formulation and a numerical algorithm for the double-gyre nonlinear shallow-water model, Numer. Math. Theor. Meth. Appl. 8 (2015) 634–650.Google Scholar
[17] Kuang, Y.Y., Wu, K.L. and Tang, H.Z., Runge-Kutta discontinuous local evolution Galerkin methods for the shallow water equations on the cubed-sphere grid, Numer. Math. Theor. Meth. Appl. 10 (2017) 373–419.Google Scholar
[18] Lele, S.A., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103(1) (1992) 16–42.Google Scholar
[19] LeVeque, R.J., Balancing source terms and flux gradients on high-resolution Godunov methods: the quasi-steady wave propagation algorithm, J. Comput. Phys. 146 (1998) 346–365.Google Scholar
[20] Liu, X.L., Zhang, S.H., Zhang, H.X. and Shu, C.-W., A new class of central compact schemes with spectral-like resolution I: Linear schemes, J. Comput. Phys. 248 (2013) 235–256.Google Scholar
[21] Li, G., Lu, C.N. and Qiu, J.X., Hybrid well-balanced WENO schemes with different indicators for shallow water equations, J. Sci. Comput. 51 (2012) 527–559.CrossRefGoogle Scholar
[22] Nonomura, T. and Fujii, K., Effects of difference scheme type in high-order weighted compact nonlinear schemes, J. Comput. Phys. 228 (2009) 3533–3539.CrossRefGoogle Scholar
[23] Nonomura, T. and Fujii, K., Robust explicit formulation of weighted compact nonlinear scheme, Comput. Fluids 85 (2013) 8–18.CrossRefGoogle Scholar
[24] Nonomura, T., Iizuka, N. and Fujii, K., Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids, Comput. Fluids 39(2) (2010) 197–214.CrossRefGoogle Scholar
[25] Nonomura, T., Iizuka, N. and Fujii, K., Increasing order of accuracy of weighted compact nonlinear scheme, In: AIAA-2007-893, 2007.Google Scholar
[26] Ren, Y.X., Liu, M. and Zhang, H., A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws, J. Comput. Phys. 192 (2003) 365–386.CrossRefGoogle Scholar
[27] Rogers, B.D., Borthwick, A.G.L. and Taylor, P.H., Mathematical balancing of flux gradient and source terms prior to using Roes approximate Riemann solver, J. Comput. Phys. 192 (2003) 422–451.CrossRefGoogle Scholar
[28] Shu, C.-W., Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: Cockburn, B., Johnson, C., Shu, C.-W., Tadmor, E. (Ed.: Quarteroni, A.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, vol. 1697, Springer, 1998, pp. 325–432.Google Scholar
[29] Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys. 77 (1988) 439–471.CrossRefGoogle Scholar
[30] Stecca, G., Siviglia, A. and Toro, E. F., A Finite Volume Upwind-Biased Centred Scheme for Hyperbolic Systems of Conservation Laws: Application to Shallow Water Equations, Commun. Comput. Phys. 12 (2012) 1183–1214.Google Scholar
[31] Vukovic, S., Crnjaric-Zic, N. and Sopta, L., WENO schemes for balance laws with spatially varying flux, J. Comput. Phys. 199 (2004) 87–109.Google Scholar
[32] Vukovic, S. and Sopta, L., ENO and WENO schemes with the exact conservation property for one-dimensional shallow water equations, J. Comput. Phys. 179 (2002) 593–621.Google Scholar
[33] Xing, Y. and Shu, C.-W., A survey of high order schemes for the shallow water equations, J. Math. Study 47 (2014) 221–249.Google Scholar
[34] Xing, Y. and Shu, C.-W., High order finite difference WENO schemes with the exact conservation property for the shallow water equations, J. Comput. Phys. 208 (2005) 206–227.Google Scholar
[35] Xing, Y. and Zhang, X., Positivity-preserving well-balanced discontinuous Galerkin methods for the shallow water equations on unstructured triangular meshes, J. Sci. Comput. 57 (2013) 19–41.CrossRefGoogle Scholar
[36] Xing, Y., Zhang, X. and Shu, C.-W., Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations, Adv. Water Resour. 33 (2010) 1476–1493.Google Scholar
[37] Zhang, S., Jiang, S. and Shu, C.-W., Development of nonlinear weighted compact schemes with increasingly higher order accuracy, J. Comput. Phys. 227 (2008) 7294–321.CrossRefGoogle Scholar
[38] Zhou, J.G., Causon, D.M., Mingham, C.G. and Ingram, D.M., The surface gradient method for the treatment of source terms in the shallow-water equations, J. Comput. Phys. 168 (2001) 1–25.Google Scholar
[39] Zhu, Q.Q., Gao, Z., Don, W.S. and Lv, X.Q., Well-balanced Hybrid Compact-WENO Schemes for Shallow Water Equations, Appl. Num. Math. 112 (2017) 65–78.CrossRefGoogle Scholar