双曲守恒律方程的高阶半拉格朗日方法

发布时间：2019-05-21 20:58

【摘要】：半拉格朗日(Semi-Lagrangian)方法广泛用来计算Vlasov方程和模拟天气预报业务,此方法将拉格朗日方法(Lagrangian)和欧拉方法(Eulerian)有效地融合在一起,同时具备了这两种方法的优点:一方面,经过改进,Semi-Lagrangian方法可以具有高阶精度;另一方面,Semi-Lagrangian方法不必受CFL条件的限制,在数值模拟时可以大量节省计算的时间。此外,加权本质无振荡格式(WENO)作为一种具有高阶精度的方法,同时具有无振荡的性质。正是由于高阶Semi-Lagrangian方法既能达到高阶精确,又能有效地处理振荡,本文针对守恒律方程提出了几种高阶的Semi-Lagrangian方法,并且通过数值模拟实验检验了方法的高阶精度和无振荡的性质,进一步丰富了Semi-Lagrangian方法求解守恒律方程的理论知识。首先,提出了一维双曲守恒律方程的高阶Semi-Lagrangian有限体积(FV)方法。采用向左的4阶RK方法计算特征曲线的初值问题,并利用特征曲线进行不同时间层的函数值的等价转换,转换后的函数值可由WENO方法重构来增加空间上的精度。由于沿着特征曲线的轨迹,初始点和终点的位置关系是变化的,故文中给出了适用于不同情形的WENO重构。进一步,通过精度检测和对无振荡性质的分析,验证了算法的高精度和有效捕捉间断点的性质。其次,提出了二维双曲守恒律方程的高阶Semi-Lagrangian有限差分(FD)方法。利用Legendre多项式构造了新的WENO方法,该方法与普通的WENO格式有同样的模板和精度,但不需采用积分计算去实现整个重构过程,节省了计算时间,更适合重构文中的不在网格点上的数值流通量。除此之外,文中给出的一系列二维守恒律方程的数值试验,验证了该方法的高阶精度和处理间断点的能力。最后,提出了5阶映射紧致Semi-Lagrangian FD方法。根据特征速度的符号,本文构造了不同的WENO重构方法,并将方法做了推广。由于采用常见的非线性权的WENO方法会造成极值点附近的精度下降,因此本文介绍了映射的加权处理这类问题。在数值模拟时,利用精度分析和无振荡性质的分析,验证了5阶映射紧致Semi-Lagrangian FD方法能达到5阶精度,同时能维持捕捉间断点的能力。综上所述,基于高阶Semi-Lagrangian方法在求解双曲守恒律方程时的高阶精度及高分辨率等特性,本文依次提出了一维标量方程、Euler方程和带源项浅水方程的高阶Semi-Lagrangian FV方法,二维双曲守恒律方程的高阶Semi-Lagrangian FD方法以及5阶映射紧致Semi-Lagrangian FD方法。数值模拟试验证明了这些方法的精度和无振荡性,体现了高阶Semi-Lagrangian方法计算双曲守恒律方程的优越性,同时说明了本文提出的方法适用于求解双曲守恒律方程。
[Abstract]:The semi-Lagrangian (Semi-Lagrangian) method is widely used to calculate the Vlasov equation and simulate the weather forecasting operation. This method effectively combines the Lagrangian method (Lagrangian) with the Euler method (Eulerian). At the same time, it has the advantages of these two methods: on the one hand, after improvement, the Semi-Lagrangian method can have high-order accuracy; On the other hand, Semi-Lagrangian method does not need to be limited by CFL condition, and can save a lot of calculation time in numerical simulation. In addition, the weighted essential non-oscillatory scheme (WENO), as a method with high order accuracy, has the property of non-oscillatory at the same time. It is precisely because the high-order Semi-Lagrangian method can not only achieve high-order accuracy, but also effectively deal with oscillations. In this paper, several higher-order Semi-Lagrangian methods are proposed for conservation law equations. The higher order accuracy and non-oscillatory properties of the method are verified by numerical simulation experiments, which further enriches the theoretical knowledge of solving conservation law equations by Semi-Lagrangian method. Firstly, a higher order Semi-Lagrangian finite volume (FV) method for one-dimensional hyperbolic conservation law equations is proposed. The fourth order RK method to the left is used to calculate the initial value problem of the feature curve, and the equivalent transformation of the function value of different time layers is carried out by using the characteristic curve. The transformed function value can be reconstructed by WENO method to increase the spatial accuracy. Because the position relationship between the initial point and the end point is changed along the trajectory of the characteristic curve, the WENO reconstruction suitable for different cases is given in this paper. Furthermore, the high precision and effective capture of discontinuity points are verified by precision detection and analysis of non-oscillatory properties. Secondly, a high-order Semi-Lagrangian finite difference (FD) method for two-dimensional hyperbolic conservation law equations is proposed. A new WENO method is constructed by using Legendre multinomial. This method has the same template and accuracy as the ordinary WENO scheme, but it does not need integral calculation to realize the whole reconstruction process, which saves the calculation time. It is more suitable to reconstruct the numerical flux which is not on the grid point in the paper. In addition, a series of numerical experiments of two-dimensional conservation law equations are given in this paper, which verify the high-order accuracy of the method and the ability to deal with intermittent points. Finally, a 5-order mapping compact Semi-Lagrangian FD method is proposed. According to the symbol of characteristic velocity, different WENO reconstruction methods are constructed and extended. Because the common nonlinear weight WENO method will reduce the accuracy near the extreme point, this paper introduces the weighting of mapping to deal with this kind of problem. In the numerical simulation, the accuracy analysis and the analysis of non-oscillatory properties are used to verify that the 5-order mapping compact Semi-Lagrangian FD method can achieve the fifth-order accuracy and maintain the ability to capture the breakpoints at the same time. In summary, based on the high-order accuracy and high resolution of the higher-order Semi-Lagrangian method in solving hyperbolic conservation law equations, the one-dimensional scalar equation, the Euler equation and the higher-order Semi-Lagrangian FV method with source term shallow water equation are proposed in this paper. The higher order Semi-Lagrangian FD method and the fifth order mapping compact Semi-Lagrangian FD method for two dimensional hyperbolic conservation law equations. The numerical simulation results show that these methods are accurate and non-oscillatory, and show the superiority of higher order Semi-Lagrangian method in calculating hyperbolic conservation law equations. at the same time, it is shown that the method proposed in this paper is suitable for solving hyperbolic conservation law equations.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.82

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