|
|
不规则沟渠浅水流动的高阶平衡有限差分AWENO格式
|
Abstract:
在本文中，我们提出了具有不规则几何形状和非平坦底部地形的开放沟渠中浅水方程的高阶有限差分加权本质无振荡(Alternative Weighted Essentially Non-Oscillatory) AWENO格式。所提出的格式保持了静水稳态解的良好平衡性质，即在通量梯度和源项之间存在精确平衡时，在离散状态保持稳态。与具有恒定横截面的传统浅水方程相比，由于沟渠不规则几何形状引起的影响，构建良好平衡格式并不是一项简单的工作。为了保持良好平衡性质，我们首先重新构造源项，然后使用具有Lax-Friedrichs通量的静水重构方法，最后借助一种新颖的源项逼近方法离散源项。基准数值示例被应用来验证所得格式的良好性能：平衡性能，高阶精度，以及对不连续解的高分辨率。
In this paper, we present high-order finite alternative weighted essentially non-oscillatory (AWENO) schemes for shallow water flows along open channels with irregular geometry and over a non-flat bottom topography. The proposed schemes maintain the well-balanced property for the still water steady-state solutions, namely preserving steady state at the discrete level, when there is an exact balance between the flux gradient and the source term. Compared with the traditional shallow water equations with constant cross-section, the construction of the well-balanced schemes is not trivial work due to the effect induced by the irregular geometry of the channels. To preserve the well-balanced property, we first reformulate the source term, then use the hydrostatic reconstruction (HR) method with Lax-Friedrichs (LF) flux, and finally discrete the source term with the help of a novel source term approximation. Benchmark numerical examples are applied to validate the good performances of the resulting schemes: well-balanced property, high order accuracy, and high resolution for the discontinuous solutions.
| [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) Numerical Methods for Shallow-Water Flow. Springer, 15-25. https://doi.org/10.1007/978-94-015-8354-1_2 |
| [4] | Castro, M.J., García-Rodríguez, J.A., González-Vida, J.M., Macías, J., et al. Numerical Simulation of Two-Layer Shallow Water Flows Through Channels with Irregular Geometry. Journal of Computational Physics, 195, 202-235. |
| [5] | Jiang, Y., Shu, C.-W. and Zhang, M. (2013) An Alternative Formulation of Finite Difference Weighted ENO Schemes with Lax-Wendroff Time Discretization for Conservation Laws, SIAM J. Journal of Scientific Computing, 35, A1137-A1160. https://doi.org/10.1137/120889885 |
| [6] | Wang, B.-S., Li, P., Gao, Z. and Don, W.S. (2018) An Improved Fifth Order Alternative WENO-Z Finite Difference Scheme for Hyperbolic Conservation Laws. Journal of Computational Physics, 374, 469-477. https://doi.org/10.1016/j.jcp.2018.07.052 |
| [7] | Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R. and Perthame, B. (2004) A Fast and Stablewell-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows, SIAM J. Journal of Scientific Computing, 25, 2050-2065. https://doi.org/10.1137/S1064827503431090 https://doi.org/10.1137/S1064827503431090 |
| [8] | Audusse, E. and Bristeau, M.O. (2005) A Well-Balanced Positivity Preserving Second-Order Scheme for Shallow Water Flows on Unstructured Meshes. Journal of Computational Physics, 206, 311-333. https://doi.org/10.1016/j.jcp.2004.12.016 |
| [9] | 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 Computational and Applied Mathematics, 340, 546-560. https://doi.org/10.1016/j.cam.2017.10.027 |
| [10] | Xing, Y. 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 |
| [11] | Shu, C.-W. and Osher, S. (1988) Efficient Implementation of Essentially Non-Oscillatory Shock Capturing Schemes. Journal of Computational Physics, 77, 439-471. https://doi.org/10.1016/0021-9991(88)90177-5 |
| [12] | Balbas, J. and Karni, S. (2009) A Central Scheme for Shallow Water Flows Along Channels with Irregular Geometry. ESAIM: Mathematical Modelling and Numerical Analysis, 43, 333-351. https://doi.org/10.1051/m2an:2008050 |
| [13] | Xing, Y. (2016) High Order Finite Volume WENO Schemes for the Shallow Water Flows Through Channels with Irregular Geometry. Journal of Computational and Applied Mathematics, 299, 229-244. https://doi.org/10.1016/j.cam.2015.11.042 |
| [14] | Kurganov, A. and Levy, D. (2002) Central-Upwind Schemes for the Saint-Venant System. ESAIM: Mathematical Modelling and Numerical Analysis, 36, 397-425. https://doi.org/10.1051/m2an:2002019 |
| [15] | García-Navarro, P., Alcrudo, F. and Savirón, J.M. (1992) 1D Open-Channel Flow Simulation Using TVD-Mccormack Scheme. Journal of Hydraulic Engineering, 118, 1359-1372. https://doi.org/10.1061/(ASCE)0733-9429(1992)118:10(1359) |
| [16] | Vazquez-Cendon, M.E. (1999) Improved 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 |
| [17] | Xing, Y. 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 |
