|
|
浅水波方程组的熵稳定有限体积格式
|
Abstract:
本文针对非平底地形上的浅水波方程组，提出了一种高精度熵稳定有限体积格式。首先，我们构造了一个具有二阶精度的well-balanced的半离散熵守恒格式，该格式满足给定熵对的熵恒等式，并精确地保持静水稳态。本文的关键点是使通量梯度和源项的离散化相互匹配；其次，二阶熵守恒格式的可负担熵守恒通量对于最终的高阶格式也是至关重要的。然后，以二阶熵守恒格式为基本模块，实现了高阶well-balanced的半离散熵守恒格式。第三，通过在现有的熵守恒格式中添加适当的耗散项，提出了高阶精度的well-balanced的半离散熵稳定格式，该耗散项基于标度熵变量的加权本质非振荡重构，以克服熵守恒的数值振荡。最后，使用Runge-Kutta方法对半离散格式进行时间积分，以实现最终格式。用大量的数值结果说明所提出的格式满足离散熵不等式，具有良好的平衡性，保持了光滑解的真正高阶精度，并且能够很好地捕获稳态下的小扰动。
This article develops a high-order accurate entropy stable finite volume scheme for the shallow wa-ter equations over non-flat bottom topography. Firstly, we construct a second-order accurate well-balanced semi-discrete entropy conservative scheme, which satisfies the entropy identity for the given entropy pair and maintains the still water steady states exactly. The key idea is to make both discretizations for the flux gradient and the source term match each other. Secondly, afforda-ble entropy conservative fluxes of the second-order entropy conservative schemes are also critical for the ultimate high-order scheme. Then, high-order well-balanced semi-discrete entropy con-servative scheme is achieved by taking the second-order entropy conservative schemes as a build-ing block. Thirdly, the high-order accurate well-balanced semi-discrete entropy stable schemes are proposed by adding a suitable dissipation term to the existing entropy conservative scheme based on the weighted essentially non-oscillatory reconstruction of the scaled entropy variables to sup-press the numerical oscillations of the entropy conservative scheme. Finally, the semi-discrete scheme is integrated in time using the Runge-Kutta approach to achieve the eventual scheme. Ex-tensive numerical results strongly indicate that the proposed scheme satisfies the discrete entropy inequality, is well-balanced, keeps the genuine high-order accuracy for smooth solutions, and can well capture small perturbations according to the steady state.
| [1] | Bradford, S.F. and Sanders, B.F. (2002) Finite-Volume Model for Shallow Water Flooding of Arbitrary Topography. Journal of Hydraulic Engineering, 128, 289-298. https://doi.org/10.1061/(ASCE)0733-9429(2002)128:3(289) |
| [2] | Gottardi, G. and Venutelli, M. (2004) Central Scheme for the Two-Dimensional Dam-Break Flow Simulation. Advances in Water Resources, 27, 259-268. https://doi.org/10.1016/j.advwatres.2003.12.006 |
| [3] | Vreugdenhil, C.B. (1995) Equations. In: Numerical Methods for Shallow-Water Flow, Water Science and Technology Library, Vol. 13, Springer, Dordrecht, 15-46. https://doi.org/10.1007/978-94-015-8354-1_2 |
| [4] | Perthame, B. and Simeoni, C. (2001) A Kinetic Scheme for the Saint-Venant System with a Source Term. CALCOLO, 38, 201-231. https://doi.org/10.1007/s10092-001-8181-3 |
| [5] | Xu. K. (2002) A Well-Balanced Gas-Kinetic Scheme for the Shallow-Water Equations with Source Terms. Journal of Computational Physics, 178, 533-562. https://doi.org/10.1006/jcph.2002.7040 |
| [6] | Kurganov, A. and Levy, D. (2002) Central-Upwind Schemes for the Saint-Venant System. Mathematical Modelling and Numerical Analysis, 36, 397-425. https://doi.org/10.1051/m2an:2002019 |
| [7] | Vukovic, S. and Sopta, L. (2002) ENO and WENO Schemes with the Exact Conservation Property for One-Dimen- sional Shallow Water Equations. Journal of Computational Physics, 179, 593-621.
https://doi.org/10.1006/jcph.2002.7076 |
| [8] | Vukovic, S., Crnjaric-Zic, N. and Sopta, L. (2004) WENO Schemes for Balance Laws with Spatially Varying Flux. Journal of Computational Physics, 199, 87-109. https://doi.org/10.1016/j.jcp.2004.02.002 |
| [9] | Xing, Y.L. and Shu, C.-W. (2005) High Order Finite Difference WENO Schemes with the Exact Conservation Property for the Shallow Water Equations. Journal of Computational Physics, 208, 206-227.
https://doi.org/10.1016/j.jcp.2005.02.006 |
| [10] | Noelle, S., Pankratz, N., Puppo, G. and Natvig, J. (2006) Well-Balanced Finite Volume Schemes of Arbitrary Order of Accuracy for Shallow Water Flows. Journal of Computa-tional Physics, 213, 474-499.
https://doi.org/10.1016/j.jcp.2005.08.019 |
| [11] | Xing, Y.L. and Shu, C.-W. (2006) High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms. Journal of Computational Physics, 214, 567-598. https://doi.org/10.1016/j.jcp.2005.10.005 |
| [12] | Noelle, S., Xing, Y. and Shu, C.-W. (2007) High-Order Well-Balanced Finite Volume WENO Schemes for Shallow Water Equation with Moving Water. Journal of Computational Physics, 226, 29-58.
https://doi.org/10.1016/j.jcp.2007.03.031 |
| [13] | Li, G., Lu, C. and Qiu, J. (2012) Hybrid Well-Balanced WENO Schemes with Different Indicators for Shallow Water Equations. Journal of Scientific Computing, 51, 527-559. https://doi.org/10.1007/s10915-011-9520-4 |
| [14] | Li, G., Caleffi, V. and Qi, Z.K. (2015) A Well-Balanced Finite Difference WENO Scheme for Shallow Water Flow Model. Applied Mathematics and Computation, 265, 1-16. https://doi.org/10.1016/j.amc.2015.04.054 |
| [15] | Zhu, Q., Gao, Z., Don, W.S. and Lv, X. (2017) Well-Balanced Hy-brid Compact-WENO Scheme for Shallow Water Equations. Applied Numerical Mathematics, 112, 65-78. https://doi.org/10.1016/j.apnum.2016.10.001 |
| [16] | Li, J.J., Li, G., Qian, S.G. and Gao, J.M. (2021) High-Order Well-Balanced Finite Volume WENO Schemes with Conservative Variables Decomposition for Shallow Water Equa-tions. Advances in Applied Mathematics and Mechanics, 13, 827-849. https://doi.org/10.4208/aamm.OA-2020-0138 |
| [17] | Caleffi, V. (2011) A New Well-Balanced Hermite Weighted Essentially Non-Oscillatory Scheme for Shallow Water Equations. International Journal for Numerical Methods in Flu-ids, 67, 1135-1159. https://doi.org/10.1002/fld.2410 |
| [18] | Russo, G. (2000) Central Schemes for Balance Laws. In: Freistühler, H. and Warnecke, G., Eds., Hyperbolic Problems: Theory, Numerics, Applications, ISNM International Se-ries of Numerical Mathematics, Vol. 141, Birkh？user, Basel, 821-829. https://doi.org/10.1007/978-3-0348-8372-6_35 |
| [19] | Touma, R. and Khankan, S. (2012) Well-Balanced Unstag-gered Central Schemes for One and Twodimensional Shallow Water Equation Systems. Applied Mathematics and Com-putation, 218, 5948-5960.
https://doi.org/10.1016/j.amc.2011.11.059 |
| [20] | Dong, J. (2020) A Robust Central Scheme for the Shallow Water Flows with an Abrupt Topography Based on Modified Hydrostatic Reconstructions. Mathematical Methods in the Ap-plied Sciences, 43, 9024-9045.
https://doi.org/10.1002/mma.6597 |
| [21] | Dong, J. and Li, D.F. (2021) Well-Balanced Nonstaggered Central Schemes Based on Hydrostatic Reconstruction for the Shallow Water Equations with Coriolis Forces and Topography. Mathemat-ical Methods in the Applied Sciences, 44, 1358-1376. https://doi.org/10.1002/mma.6835 |
| [22] | Ern, A., Piperno, S. and Djadel, K. (2008) A Well-Balanced Runge-Kutta Discontinuous Galerkin Method for the Shallow-Water Equations with Flooding and Drying. International Journal for Numerical Methods in Fluids, 58, 1-25.
https://doi.org/10.1002/fld.1674 |
| [23] | Xing, Y. (2014) Exactly Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations with Moving Water Equilibrium. Journal of Computational Physics, 257, 536-553. https://doi.org/10.1016/j.jcp.2013.10.010 |
| [24] | Li, G., Song, L.N. and Gao. J.M. (2018) High Order Well-Balanced Discontinuous Galerkin Methods Based on Hydrostatic Reconstruction for Shallow Water Equations. Journal of Com-putational and Applied Mathematics, 340, 546-560. https://doi.org/10.1016/j.cam.2017.10.027 |
| [25] | Vignoli, G., Titarev, V.A. and Toro, E.F. (2008) ADER Schemes for the Shallow water Equations in Channel with Irregular Bottom Elevation. Journal of Computational Physics, 227, 2463-2480.
https://doi.org/10.1016/j.jcp.2007.11.006 |
| [26] | Navas-Montilla, A. and Murillo, J. (2015) Energy Balanced Numer-ical Schemes with very High Order. The Augmented Roe Flux ADER Scheme. Application to the Shallow Water Equa-tions. Journal of Computational Physics, 290, 188-218. https://doi.org/10.1016/j.jcp.2015.03.002 |
| [27] | Gassner, G.J., Winters A.R. and Kopriva, D.A. (2016) A Well Balanced and Entropy Conservative Discontinuous Galerkin Spec-tral Element Method for the Shallow Water Equations. Applied Mathematics and Computation, 272, 291-308. https://doi.org/10.1016/j.amc.2015.07.014 |
| [28] | Berthon, C. and Chalons, C. (2016) A Fully Well-Balanced, Posi-tive and Entropy-Satisfying Godunov-Type Method for the Shallow-Water Equations. Mathematics of Computation, 85, 1281-1307. https://doi.org/10.1090/mcom3045 |
| [29] | Li, D.F. and Dong, J. (2022) A Robust Hybrid Unstaggered Central and Godunov-Type Scheme for Saint-Venant-Exner Equations with Wet/Dry Fronts. Computers & Fluids, 235, Article ID: 105284.
https://doi.org/10.1016/j.compfluid.2021.105284 |
| [30] | Yuan, X.H. (2018) A Well-Balanced Element-Free Galerkin Method for the Nonlinear Shallow Water Equations. Applied Mathematics and Computation, 331, 46-53. https://doi.org/10.1016/j.amc.2018.01.061 |
| [31] | Liu, X., Mohammadian, A., Kurganov, A. and Sedano, J.A.I. (2015) Well-Balanced Central-Upwind Scheme for a Fully Coupled Shallow Water System Modeling Flows over Erodible Bed. Journal of Computational Physics, 300, 202-218. https://doi.org/10.1016/j.jcp.2015.07.043 |
| [32] | Chertock, A., Kurganov, A. and Wu, T. (2020) Operator Splitting Based Central-Upwind Schemes for Shallow Water equations with Moving Bottom Topography. Communications in Mathematical Sciences, 18, 2149-2168.
https://doi.org/10.4310/CMS.2020.v18.n8.a3 |
| [33] | Li, G., Li, J.J., Qian, S.G. and Gao, J.M. (2021) A Well-Balanced ADER Discontinuous Galerkin Method Based on Differential Transformation Procedure for Shallow Water Equations. Applied Mathematics and Computation, 395, Article ID: 125848. https://doi.org/10.1016/j.amc.2020.125848 |
| [34] | Duan, J.M. and Tang. H.Z. (2021) High-Order Accurate Entropy Stable Finite Difference Schemes for the Shallow Water Magnetohydrodynamics. Journal of Computational Physics, 431, Article ID: 110136.
https://doi.org/10.1016/j.jcp.2021.110136 |
| [35] | Winters, A.R. and Gassner, G.J. (2015) A Comparison of Two En-tropy Stable Discontinuous Galerkin Spectral Element Approximations for the Shallow Water Equations with Non-Constant Topography Author Links Open Overlay Panel. Journal of Computational Physics, 301, 357-376. https://doi.org/10.1016/j.jcp.2015.08.034 |
| [36] | Wintermeyer, N., Winters, A.R., Gassner, G.J. and Kopriva, D.A. (2017) An Entropy Stable Nodal Discontinuous Galerkin Method for the Two Dimensional Shallow Water Equations on Unstructured Curvilinear Meshes with Discontinuous Bathymetry. Journal of Computational Physics, 340, 200-242. https://doi.org/10.1016/j.jcp.2017.03.036 |
| [37] | Wintermeyer, N., Winters, A.R., Gassner, G.J. and Warburton, T. (2018) An entropy stable discontinuous Galerkin method for the Shallow Water Equations on Curvilinear Meshes with Wet/Dry Fronts Accelerated by GPUs. Journal of Computational Physics, 375, 447-480. https://doi.org/10.1016/j.jcp.2018.08.038 |
| [38] | Wen, X., Don, W.S., Gao, Z. and Xing, Y.L. (2020) Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations Journal of Scientific Computing, 83, Article No. 66.
https://doi.org/10.1007/s10915-020-01248-3 |
| [39] | Liu, Q.S., Liu, Y.Q. and Feng, J.H. (2019) The Scaled Entropy Variables Reconstruction for Entropy Stable Schemes with Application to Shallow Water Equations. Computers & Fluids, 192, Article ID: 104266.
https://doi.org/10.1016/j.compfluid.2019.104266 |
| [40] | Fjordholm, U.S., Mishra, S. and Tadmor, E. (2011) Well-Balanced and Energy Stable Schemes for the Shallow Water Equations with Discontinuous Topography Journal of Computational Physics, 230, 5587-5609.
https://doi.org/10.1016/j.jcp.2011.03.042 |
| [41] | Fjordholm, U.S., Mishra, S. and Tadmor, E. (1987) The Numerical Viscosity of Entropy Stable Schemes for Systems of Conservation Laws. I. Mathematics of Computation, 49, 91-103.
https://doi.org/10.1090/S0025-5718-1987-0890255-3 |
| [42] | Lefloch, P.G., Mercier, J.M. and Rohde, C. (2002) Fully Discrete Entropy Conservative Schemes of Arbitrary Order. SIAM Journal on Numerical Analysis, 40, 1968-1992. https://doi.org/10.1137/S003614290240069X |
| [43] | Fjordholm, U.S., Mishra, S. and Tadmor, E. (2012) Arbitrarily High-Order Accurate Entropy Stable Essentially Non- Oscillatory Schemes for Systems of Conservation Laws. SIAM Journal on Numerical Analysis, 50, 544-573.
https://doi.org/10.1137/110836961 |
| [44] | Fjordholm, U.S., Mishra, S. and Tadmor, E. (2013) ENO Reconstruction and ENO Interpolation Are Stable. Foundations of Computational Mathematics, 13, 139-159. https://doi.org/10.1007/s10208-012-9117-9 |
| [45] | Shu, C.-W. (1997) Essentially Non-Oscillatory and Weighted Es-sentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws. NASA/CR-97-206253, ICASE Report No. 97-65. |
| [46] | Shu, C.-W. (2009) High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems. SIAM Review, 51, 82-126. https://doi.org/10.1137/070679065 |
| [47] | Shu, C.-W. (2016) High Order WENO and DG Methods for Time-Dependent Convectiondominated PDEs: Abrief Survey of Several Recent Develop-ments. Journal of Computational Physics, 316, 598-613.
https://doi.org/10.1016/j.jcp.2016.04.030 |
| [48] | Biswas, B. and Dubey, R.K. (2018) Low Dissipative Entropy Stable Schemes Using Third Order WENO and TVD Reconstructions. Advances in Computational Mathematics, 44, 1153-1181. https://doi.org/10.1007/s10444-017-9576-2 |
| [49] | Shu, C.-W. and Osher, S. (1998) Efficient Implemen-tation of Essentially Non-Oscillatory Shockcapturing Schemes. Journal of Computational Physics, 77, 439-471. https://doi.org/10.1016/0021-9991(88)90177-5 |
| [50] | LeVeque, R.J. (1998) Balancing Source Terms and Flux Gra-dient on High-Resolution Godunov Methods: The Quasi-Steady Wave-Propagation Algorithm. Journal of Computation-al Physics, 146, 346-365.
https://doi.org/10.1006/jcph.1998.6058 |
| [51] | Xing, Y.L. and Shu, C.-W. (2006) A New Approach of High Order Well-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms. Communications in Computational Physics, 1, 100-134. |
| [52] | Vazquez-Cendon, M.E. (1999) Im-proved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry. Journal of Computational Physics, 148, 497-526.
https://doi.org/10.1006/jcph.1998.6127 |
| [53] | Goutal, N. and Maurel, F. (1997) Proceedings of the Second Work-shop on Dam-Break Wave Simulation. Technical Report HE-43/97/016/A, Electricité de France, Département Laboratoire National d’Hydraulique, Groupe Hydraulique Fluviale. |
| [54] | Bermudez, A. and Vazquez, M.E. (1994) Upwind Methods for Hyperbolic Conservation Laws with Source Terms. Computers & Fluids, 23, 1049-1071. https://doi.org/10.1016/0045-7930(94)90004-3 |
| [55] | Alcrudo, F. and Benkhaldoun, F. (2001) Exact Solution to the Riemann Problem of the Shallow Water Euqaionts with a Bottom Step. Computers & Fluids, 30, 643-671. https://doi.org/10.1016/S0045-7930(01)00013-5 |
