浅水方程高分辨率有限差分方法研究
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摘要
公路交通污染主要是汽车尾气对大气污染和个别交通事故对江河等水环境污染。将浅水方程和污染物迁移方程耦合，可以模拟由交通污染事故所导致的有毒有害物质在重要水域的迁移过程，对下游水质进行预测，为交通污染事故应急处置提供指导。浅水方程在水利、海岸、海洋和环境工程等领域都具有重要的应用。处理间断是数值求解浅水方程的关键之一，也是困难所在。高分辨率方法就是为更好地处理间断问题而设计的。本文重点研究浅水方程的高分辨率有限差分方法，并在交通污染事故应急处置中进行了分析应用。论文的主要工作包括以下几个方面：
1．提出了一种具有四阶精度的松弛格式。对一维问题，该格式以四阶中心WENO重构为基础；对二维问题，用逐维计算的方法将四阶中心WENO重构进行了推广。时间的离散采用具有较大稳定区域的Runge-Kutta类解算器Rock4。该格式保持了松弛格式简单的优点，即不用求解Riemann问题和计算通量函数的雅可比矩阵。运用本文格式对一维、二维浅水方程，以及一维气动力学Euler方程组和二维Burgers方程进行了大量的数值试验，并与三阶松弛格式的计算结果进行了比较，结果表明本文所提出的格式具有更低的数值耗散和更高的分辨率。
2．以三阶WENO重构和三阶显隐式Runge-Kutta方法为基础，提出了一种新的三阶松弛格式，其形式更简单、计算量更小。通过数值算例对新的三阶松弛格式和原三阶格式进行了比较研究。
3．将二维全离散中心格式推广于浅水方程，给出了一种计算二维浅水方程的高分辨率数值方法。应用该方法对圆柱溃坝等问题进行了数值模拟，计算结果与用其它方法所得结果吻合，表明了方法的有效性和稳定性。通过数值试验对不同限制器在数值性能上的差异进行了比较，据此确定了较优的限制器，为二维浅水方程的计算提供了良好的数值手段。
4．将三阶和五阶WENO重构方法和半离散中心迎风通量相结合，给出了计算一维浅水方程的两种高分辨率、高精度数值方法。对底坡项的离散保证了计算格式的和谐性，离散摩阻项的方法简单有效。用一维浅水方程的典型算例验证了方法的有效性和和谐性。
5．以三阶紧凑中心WENO重构方法为基础，提出了一种求解双曲型守恒律组的三阶半离散中心迎风格式，并将该格式推广于浅水方程，建立了计算二维浅水方程的和谐方法。数值算例的结果表明该方法健全、通用且具有很高的
The impact of highway traffic for environment is mainly induced by vehicle emissions on atmospheric and pollution traffic accidents for water quality of a river. A model for coupling the shallow water equations with transport equation can be used to simulate the transport of toxic and hazardous materials in water regions and predict water quality caused by traffic pollution accidents on freeway. It can provide the guidance for dealing with traffic pollution accidents. The shallow water equations have important applications in hydraulic, coastal ocean and environmental engineering. A key and difficult problem of numerical methods for such equations is that they can resolve discontinuities. The high-resolution numerical methods are designed for problems containing discontinuities. This thesis concerns with the high-resolution finite-difference methods for the shallow water equations. The application of the present method to dealing with traffic pollution accidents is discussed. The most distinctive parts can be described as following:
1. A fourth-order relaxation scheme for the shallow water equations is proposed. The scheme is based on central weighted essentially non-oscillatory(WENO) reconstruction for one space dimension. In the two-dimensional cases, this reconstruction is generalized by dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver R0CK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The presented scheme is tested on a variety of numerical experiments with one-dimensional Euler equations of gas dynamics, the two-dimensional Burgers equation and the shallow water equation in both one and two dimensions. To illustrate the improvement of our method, the results are compared with numerical solutions computed by the third-order relaxation scheme. The numerical experiments demonstrate that the present method has the higher shock resolution and smaller numerical dissipation than the third-order relaxation scheme.
2. A new third-order relaxation scheme is proposed. The scheme combines with third-order WENO reconstruction for spatial discretization and third-order implicit-explicit method for time discretization. The new scheme is much simpler and less computationally expensive than the original one. The new scheme is test on
1．提出了一种具有四阶精度的松弛格式。对一维问题，该格式以四阶中心WENO重构为基础；对二维问题，用逐维计算的方法将四阶中心WENO重构进行了推广。时间的离散采用具有较大稳定区域的Runge-Kutta类解算器Rock4。该格式保持了松弛格式简单的优点，即不用求解Riemann问题和计算通量函数的雅可比矩阵。运用本文格式对一维、二维浅水方程，以及一维气动力学Euler方程组和二维Burgers方程进行了大量的数值试验，并与三阶松弛格式的计算结果进行了比较，结果表明本文所提出的格式具有更低的数值耗散和更高的分辨率。
2．以三阶WENO重构和三阶显隐式Runge-Kutta方法为基础，提出了一种新的三阶松弛格式，其形式更简单、计算量更小。通过数值算例对新的三阶松弛格式和原三阶格式进行了比较研究。
3．将二维全离散中心格式推广于浅水方程，给出了一种计算二维浅水方程的高分辨率数值方法。应用该方法对圆柱溃坝等问题进行了数值模拟，计算结果与用其它方法所得结果吻合，表明了方法的有效性和稳定性。通过数值试验对不同限制器在数值性能上的差异进行了比较，据此确定了较优的限制器，为二维浅水方程的计算提供了良好的数值手段。
4．将三阶和五阶WENO重构方法和半离散中心迎风通量相结合，给出了计算一维浅水方程的两种高分辨率、高精度数值方法。对底坡项的离散保证了计算格式的和谐性，离散摩阻项的方法简单有效。用一维浅水方程的典型算例验证了方法的有效性和和谐性。
5．以三阶紧凑中心WENO重构方法为基础，提出了一种求解双曲型守恒律组的三阶半离散中心迎风格式，并将该格式推广于浅水方程，建立了计算二维浅水方程的和谐方法。数值算例的结果表明该方法健全、通用且具有很高的
The impact of highway traffic for environment is mainly induced by vehicle emissions on atmospheric and pollution traffic accidents for water quality of a river. A model for coupling the shallow water equations with transport equation can be used to simulate the transport of toxic and hazardous materials in water regions and predict water quality caused by traffic pollution accidents on freeway. It can provide the guidance for dealing with traffic pollution accidents. The shallow water equations have important applications in hydraulic, coastal ocean and environmental engineering. A key and difficult problem of numerical methods for such equations is that they can resolve discontinuities. The high-resolution numerical methods are designed for problems containing discontinuities. This thesis concerns with the high-resolution finite-difference methods for the shallow water equations. The application of the present method to dealing with traffic pollution accidents is discussed. The most distinctive parts can be described as following:
1. A fourth-order relaxation scheme for the shallow water equations is proposed. The scheme is based on central weighted essentially non-oscillatory(WENO) reconstruction for one space dimension. In the two-dimensional cases, this reconstruction is generalized by dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver R0CK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The presented scheme is tested on a variety of numerical experiments with one-dimensional Euler equations of gas dynamics, the two-dimensional Burgers equation and the shallow water equation in both one and two dimensions. To illustrate the improvement of our method, the results are compared with numerical solutions computed by the third-order relaxation scheme. The numerical experiments demonstrate that the present method has the higher shock resolution and smaller numerical dissipation than the third-order relaxation scheme.
2. A new third-order relaxation scheme is proposed. The scheme combines with third-order WENO reconstruction for spatial discretization and third-order implicit-explicit method for time discretization. The new scheme is much simpler and less computationally expensive than the original one. The new scheme is test on
引文
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