求解二维浅水波方程的旋转混合格式

郑素佩; 李霄; 赵青宇; 封建湖

doi:10.21656/1000-0887.420063

求解二维浅水波方程的旋转混合格式

doi: 10.21656/1000-0887.420063

长安大学理学院，西安 710064

基金项目: 国家自然科学基金（面上项目） (11971075)；陕西省自然科学基金(2020JQ-338；2019JM-243)

详细信息

作者简介:
郑素佩（1978—），女，副教授，博士，硕士生导师（E-mail：zsp2008@chd.edu.cn）

李霄（1998—），女，硕士生（通讯作者. E-mail：1012124794@qq.com）

中图分类号: O354
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- 被引次数: 0
出版历程
- 收稿日期: 2021-03-08
- 修回日期: 2021-09-27
- 网络出版日期: 2022-01-07
- 刊出日期: 2022-02-01

A Rotated Mixed Scheme for Solving 2D Shallow Water Equations

School of Sciences, Chang’an University, Xi’an 710064, P.R.China

摘要

摘要:
针对二维浅水波方程数值求解问题，构造了一种旋转通量混合格式。空间方向上，该算法利用浅水波方程通量函数的旋转不变性，在单元界面法线方向及单元界面切线方向上采用可消除红斑现象的HLL与满足热力学第二定律的熵稳定加权混合数值通量函数，时间方向上采用三阶强稳定Runge-Kutta法。数值结果表明，该混合格式对于二维浅水波方程数值求解具有分辨率高的良好特性。
- 旋转不变性 /
- 熵稳定格式 /
- HLL格式 /
- 有限体积法 /
- Runge-Kutta 法
Abstract:
A rotated flux mixed scheme was proposed for solving 2D shallow water equations. Spatially, the algorithm uses the rotation invariance of the shallow water equations. In the normal direction and tangent direction of the element interface, both the HLL, which can eliminate the carbuncle, and the entropy stable weighted hybrid numerical flux function satisfying the 2nd law of thermodynamics, were applied to give fine numerical results. Temporally, the 3rd-order strongly stable Runge-Kutta method was used. The numerical results show that, the new scheme has high resolution for solving 2D shallow water equations.
- rotation invariance /
- entropy stable scheme /
- HLL scheme /
- finite-volume method /
- Runge-Kutta method

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图 1 圆柱溃坝模拟图

Figure 1. The simulation diagram for the cylindrical dam

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图 2 二维圆柱溃坝问题密度等值线图：(a) 非混合格式密度结果；(b) 混合格式密度结果

Figure 2. The density contour for the 2D cylindrical dam-break problem：(a) the non-mixed scheme density results; (b) the mixed scheme density results

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图 3 二维圆柱溃坝问题速度图：(a) 非混合格式x方向的速度u；(b) 混合格式x方向的速度u；(c) 非混合格式y方向的速度v；(d) 混合格式y方向的速度v

Figure 3. The velocity contour for the 2D cylindrical dam-break problem: (a) the non-mixed scheme x direction velocity u results; (b) the mixed scheme x direction velocity u results; (c) the non-mixed scheme y direction velocity v results; (d) the mixed scheme y direction velocity v results

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图 4 圆形溃坝模拟图

Figure 4. The simulation diagram for the circular dam break

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图 5 二维圆形溃坝问题密度等值线图：(a) 非混合格式密度结果； (b) 混合格式密度结果

Figure 5. The density contour for the 2D circular dam-break problem: (a) the non-mixed scheme density results; (b) the mixed scheme density results

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图 6 二维圆形溃坝问题速度图：(a) 非混合格式x方向的速度u；(b) 混合格式x方向的速度u；(c) 非混合格式y方向的速度v；(d) 混合格式y方向的速度v

Figure 6. The velocity contour for the 2D circular dam-break problem: (a) the non-mixed scheme x direction velocity u results; (b) the mixed scheme x direction velocity u results; (c) the non-mixed scheme y direction velocity v results; (d) the mixed scheme y direction velocity v results

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图 7 激波聚焦模拟图

Figure 7. The simulation diagram for the shock wave focusing problem

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图 8 二维激波聚焦问题密度等值线图：(a) 非混合格式密度结果； (b) 混合格式密度结果

Figure 8. The density contour for the 2D shock wave focusing problem: (a) the non-mixed scheme density results; (b) the mixed scheme density results

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图 9 二维激波聚焦问题速度图：(a) 非混合格式x方向的速度u； (b) 混合格式x方向的速度u；(c) 非混合格式y方向的速度v ； (d) 混合格式y方向的速度v

Figure 9. The velocity contour for the 2D shock wave focusing problem: (a) the non-mixed scheme x direction velocity u results; (b) the mixed scheme x direction velocity u results; (c) the non-mixed scheme y direction velocity v results; (d) the mixed scheme y direction velocity v results

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图 10 潮汐问题模拟图

Figure 10. The simulation diagram for the tidal problem

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图 11 二维潮汐问题密度等值线图：(a) 非混合格式密度结果；(b) 混合格式密度结果

Figure 11. The density contour map for the 2D tidal problem: (a) the non-mixed scheme density results; (b) the mixed scheme density results

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图 12 二维潮汐问题速度图：(a) 非混合格式x方向的速度u；(b) 混合格式x方向的速度u；(c) 非混合格式y方向的速度v；(d) 混合格式y方向的速度v

Figure 12. The velocity contour for the 2D tidal problem: (a) the non-mixed scheme x direction velocity u results; (b) the mixed scheme x direction velocity v results; (c) the non-mixed scheme y direction velocity v results; (d) the mixed scheme y=direction velocity v results

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