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High Order Well-Balanced Weighted Compact Nonlinear Schemes for the Gas Dynamic Equations under Gravitational Fields

Part of: Hyperbolic equations and systems

Published online by Cambridge University Press:  31 January 2018

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 Province, China
*
*Corresponding author. Email addresses:[email protected] (Z. Gao), [email protected] (G. H. Hu)
*Corresponding author. Email addresses:[email protected] (Z. Gao), [email protected] (G. H. Hu)
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Abstract

In this study, we propose a high order well-balanced weighted compact nonlinear (WCN) scheme for the gas dynamic equations under gravitational fields. The proposed scheme is an extension of the high order WCN schemes developed in (S. Zhang, S. Jiang, C.-W Shu, J. Comput. Phys. 227 (2008) 7294-7321). For the purpose of maintaining the exact steady state solution, the well-balanced technique in (Y. Xing, C.-W Shu, J. Sci. Comput. 54 (2013) 645-662) is employed to split the source term into two terms. The proposed scheme can maintain the isothermal equilibrium solution exactly, genuine high order accuracy and resolve small perturbations of the hydrostatic balance state on the coarse meshes. Furthermore, 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 schemes with local characteristic projections are used to construct different WCN schemes. Several representative one- and two-dimensional examples are simulated to demonstrate the good performance of the proposed schemes.

Keywords

Euler equationsgravitational fieldssource termsteady state solutionweighted compact nonlinear Scheme

MSC classification

Secondary: 35L65: Conservation laws 35L67: Shocks and singularities
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 697 - 713
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R. and Perthame, B., A fast and stable wellbalanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. Comput. 25 (2004) 2050–2065.Google 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 nonoscillatory scheme for hyperbolic conservation laws, J. Comput. Phys. 227 (2008) 3101–3211.Google Scholar
[4] Botta, N., Klein, R., Langenberg, S. and S. Lützenkirchen, Well-balanced finite volume methods for nearly hydrostatic flows, J. Comput. Phys. 196 (2004) 539–565.Google Scholar
[5] 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
[6] Chandrashekar, P. and Zenk, M., Well-balanced nodal discontinuous Galerkin method for Euler equations with gravity, J. Sci. Comput. 71 (2015) 1062–1093.Google Scholar
[7] Chertock, A., Cui, S., Kurganovz, A., Özcan, S.N. and Tadmor, E., Well-balanced central-upwind schemes for the Euler equations with gravitation, SIAM J. Sci. Comput. submitted.Google Scholar
[8] 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
[9] Deng, X. and Maekawa, H., Compact high-order accurate nonlinear schemes, J. Comput. Phys. 130 (1997) 77–91.Google Scholar
[10] Deng, X. and Zhang, H., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys. 165 (2000) 22–44.Google Scholar
[11] Desveaux, V., Zenk, M., Berthon, C. and Klingenberg, C., A well-balanced scheme for the Euler equation with a gravitational potential. Finite Vol. Complex Appl. VII-Methods Theor. Asp. Springer Proc. Math. Stat. 77 (2014) 217–226.Google Scholar
[12] Gao, Z. and Hu, G.H., High order well-balanced weighted compact nonlinear schemes for the shallow water equations, Comm. Comput. Phys. Accepted.Google Scholar
[13] Jiang, G.S. and Shu, C.-W., Efficient implementation of weighted ENO Schemes, J. Comput. Phys. 126 (1996) 202–228.Google Scholar
[14] Käppeli, R. and Mishra, S., Well-balanced schemes for the euler equations with gravitation, J. Comput. Phys. 259 (2014) 199–219.Google Scholar
[15] Lele, S.A., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys. 103(1) (1992) 16–42.Google Scholar
[16] LeVeque, R.J., Balancing source terms and flux gradients on high-resolution Godunovmethods: the quasi-steady wave propagation algorithm, J. Comput. Phys. 146 (1998) 346–365.Google Scholar
[17] LeVeque, R.J., Bale, D.S., Wave propagation methods for conservation laws with source terms, In: Proceedings of the 7th International Conference on Hyperbolic Problems, (1998) 609–618.Google Scholar
[18] Li, G. and Xing, Y.L., Well-balanced discontinuous Galerkin Methods for the Euler equations under gravitational fields, J. Sci. Comput. 67 (2016) 493–513.Google Scholar
[19] Li, G. and Xing, Y.L., High order finite volume WENO schemes for the Euler equations under gravitational fields, J. Comput. Phys. 316 (2016) 145–163.Google Scholar
[20] 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.Google Scholar
[21] 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
[22] Luo, J., Xu, K. and Liu, N., A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field, SIAM J. Sci. Comput. 33 (2011) 2356–2381.Google Scholar
[23] Nonomura, T. and Fujii, K., Effects of difference scheme type in high-order weighted compact nonlinear schemes, J. Comput. Phys. 228 (2009) 3533–3539.Google Scholar
[24] Nonomura, T. and Fujii, K., Robust explicit formulation of weighted compact nonlinear scheme, Comput. Fluids 85 (2013) 8–18.Google Scholar
[25] Nonomura, T., Iizuka, N. and Fujii, K., Freestream and vortex preservation properties of highorder WENO and WCNS on curvilinear grids, Comput. Fluids 39(2) (2010) 197–214.Google Scholar
[26] Nonomura, T., Iizuka, N. and Fujii, K., Increasing order of accuracy of weighted compact nonlinear scheme, In: AIAA-2007-893, 2007.Google Scholar
[27] Perthame, B. and Simeoni, C., A kinetic scheme for the Saint-Venant system with a source term, Calcolo 38, (2001) 201–231.Google 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.: A. Quarteroni) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics, Springer, 1697 (1998) 325–432.Google Scholar
[29] Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shockcapturing schemes, J. Comput. Phys. 77 (1988) 439–471.Google Scholar
[30] Slyz, A., Prendergast, K.H., Time independent gravitational fields in the BGK scheme for hydrodynamics, Astron. Astrophys. Suppl. Ser. 139 (1999) 199–217.Google Scholar
[31] Tian, C.T., Xu, K., Chan, K.L. and Deng, L.C., A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields, J. Comput. Phys. 226 (2007) 2003–2027.Google Scholar
[32] 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
[33] Xing, Y. and Shu, C.-W., High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields, J. Sci. Comput. 54 (2013) 645–662.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 Shu, C.-W., High order well-balanced finite difference WENO schemes for a class of hyperbolic systems with source terms, J. Sci. Comput. 27 (2006) 477–494.Google Scholar
[36] 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.Google Scholar
[37] Xu, K., A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J. Comput. Phys. 178 (2002) 533–562.Google Scholar
[38] 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.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.Google Scholar