基于特征思想的高分辨率格式的研究和应用
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摘要
本文为数值求解依赖时间的偏微分方程提出两类基于特征思想的高分辨率格式。从而我们主要考虑两部分内容。首先基于CIP方法和高阶紧致方法,我们提出一种新型特征插值高阶紧致差分方法,并将其运用于数值模拟非线性波动方程。然后我们对守恒型双曲方程提出一种基于特征思想的有限体积方法,并应用于数值求解Eulelr方程组和浅水方程组。
论文的第一大部分,我们对非线性波动方程提出一种非守恒Semi-Lagrangian方法。这种格式的主要构造思想如下:根据两相邻节点的节点值和导数值构造三次多项式,然后运用Semi-Lagrangian的方法求解节点沿特征线移动的位置,由此三次多项式和节点位置得到时间上的推进。不同于传统的CIP格式,我们利用最初由Lele提出的高阶紧致格式直接利用节点值求解一阶导数值。最后我们将这种格式运用于数值模拟Burgers方程和KdV方程验证格式的高分辨率的性质。通过一系列的数值试验验证格式的有效性。最后将这种格式推广到一维的Euler方程组。
论文的第二大部分,我们对双曲守恒方程组构造有限体积格式,由于Euler方程组在空气动力学中的的重要性,我们主要基于Euler方程组构造格式。这种格式的空间导数通过有限体积离散,时间上采用Simpson公式离散。其中有限体积格式中的通量值通过单元边界点的值得到,而该单元边界点的值是通过中心加权重构得到。求解此节点值的主要思想如下:首先运用三阶或四阶Runge-Kutta方法求解特征方程,从而得到节点沿特征线的位置。然后采用CWENO重构得到多项式,我们分别构造三阶和五阶CWENO格式构造多项式,最后利用节点位置和此多项式得到节点值。这种高分辨率的有限体积格式结合特征思想和中心加权格式的优点。然后我们将这种基于特征思想的有限体积格式应用于一维的标量和Euler守恒方程,通过经典的算例验证格式的高分率和收敛的性质。
最后将这种基于特征思想的高阶有限体积格式应用于一些浅水问题,从而验证格式,如一维溃坝问题,临界左稀疏波问题,两稀疏波中间几乎为干底问题,干溃坝问题,生成干溃坝问题等。另外,我们同时将这种格式按照分裂方法推广到二维守恒律情形。并通过对二维的溃坝问题的应用验证格式的有效性,通过与WAF格式以及其它高阶数值格式相比较可以看到该格式得到非常好的数值结果。
This dissertation introduces two types of characteristic-based high order numerical methods for solutions of time-dependent nonlinear partial differential equations, and thus it is composed of two main parts.First,a new finite difference method,which combines Constrained Interpolation Profile(CIP) method and the High-order compact (HOC) method,is proposed to numerically solve the wave propagation problems of nonlinear evolution equations.Second,we discuss a characteristic-based finite volume method for hyperbolic conservation laws,in particular we use it to solve the Euler equations and shallow water equations.
In the first part of dissertation,a non-conservative semi-Lagrangian scheme,the CIP-HOC coupling method,is presented for solving some nonlinear waves equations. The main idea of the proposed scheme is as follows.A third-order polynomial is constructed through interpolating the points and derivative values at two adjacent grid points.Unlike to the traditional CIP method,the derivative used in the presented scheme is obtained by using a high-order compact scheme originally proposed by Lele. The position of grid point along the characteristic curves can be obtained by using the semi-Lagrangian method.The evolution of the point value at the next time step is defined by using the obtained polynomial at the position of the grid point.To test the high accurate property of the scheme,we applied to solve the Burger's and KdV equations. A series of numerical experiments are given,and numerical results also verify the effectiveness of the new scheme.In the end,this scheme is also extended to solve one dimensional Euler equations.
In the second part,we discuss a characteristic-based finite volume method for hyperbolic conservation laws.We focus on the Euler system due to its importance in gas dynamics.In the scheme,the spatial derivatives are discritized by a finite volume method,while in time,the Simpson's quadrature rule is used.The point values at the cell boundaries are obtained by using the Central WENO reconstruction.The method is composed of the following steps:the position of the grid points along the characteristic curves are computed by using the third or fourth order Runge-Kutta method,while the polynomial is obtained by using the third order or fifth order CWENO reconstruction. Then,we obtain the grid point values by using the polynomial function at the point position.Finally,the numerical flux in the new finite volume scheme is obtained based on the point value.The new high resolution finite volume method is combined with the high resolution property of the characteristic method and the non-oscillatory property of the CWENO method.Some classical tests for both scalar and Euler conservations laws in one dimension are performed to verify he accuracy and convergence of the present scheme.
In the end,some benchmark tests of shallow wave equations are also adopted to verify the presented finite volume scheme,such as 1D dam-break with large depth difference,left critical rarefaction,two rarefactions and nearly dry bed,dam-break problem with dry bed and the generation of dry bed in the middle.Furthermore,we also extend the scheme to 2D conservation laws by using the dimensional splitting approach.For the 2D dam-break problems,the performance of the present scheme has been compared with the WAF scheme and other high-order schemes.
论文的第一大部分,我们对非线性波动方程提出一种非守恒Semi-Lagrangian方法。这种格式的主要构造思想如下:根据两相邻节点的节点值和导数值构造三次多项式,然后运用Semi-Lagrangian的方法求解节点沿特征线移动的位置,由此三次多项式和节点位置得到时间上的推进。不同于传统的CIP格式,我们利用最初由Lele提出的高阶紧致格式直接利用节点值求解一阶导数值。最后我们将这种格式运用于数值模拟Burgers方程和KdV方程验证格式的高分辨率的性质。通过一系列的数值试验验证格式的有效性。最后将这种格式推广到一维的Euler方程组。
论文的第二大部分,我们对双曲守恒方程组构造有限体积格式,由于Euler方程组在空气动力学中的的重要性,我们主要基于Euler方程组构造格式。这种格式的空间导数通过有限体积离散,时间上采用Simpson公式离散。其中有限体积格式中的通量值通过单元边界点的值得到,而该单元边界点的值是通过中心加权重构得到。求解此节点值的主要思想如下:首先运用三阶或四阶Runge-Kutta方法求解特征方程,从而得到节点沿特征线的位置。然后采用CWENO重构得到多项式,我们分别构造三阶和五阶CWENO格式构造多项式,最后利用节点位置和此多项式得到节点值。这种高分辨率的有限体积格式结合特征思想和中心加权格式的优点。然后我们将这种基于特征思想的有限体积格式应用于一维的标量和Euler守恒方程,通过经典的算例验证格式的高分率和收敛的性质。
最后将这种基于特征思想的高阶有限体积格式应用于一些浅水问题,从而验证格式,如一维溃坝问题,临界左稀疏波问题,两稀疏波中间几乎为干底问题,干溃坝问题,生成干溃坝问题等。另外,我们同时将这种格式按照分裂方法推广到二维守恒律情形。并通过对二维的溃坝问题的应用验证格式的有效性,通过与WAF格式以及其它高阶数值格式相比较可以看到该格式得到非常好的数值结果。
This dissertation introduces two types of characteristic-based high order numerical methods for solutions of time-dependent nonlinear partial differential equations, and thus it is composed of two main parts.First,a new finite difference method,which combines Constrained Interpolation Profile(CIP) method and the High-order compact (HOC) method,is proposed to numerically solve the wave propagation problems of nonlinear evolution equations.Second,we discuss a characteristic-based finite volume method for hyperbolic conservation laws,in particular we use it to solve the Euler equations and shallow water equations.
In the first part of dissertation,a non-conservative semi-Lagrangian scheme,the CIP-HOC coupling method,is presented for solving some nonlinear waves equations. The main idea of the proposed scheme is as follows.A third-order polynomial is constructed through interpolating the points and derivative values at two adjacent grid points.Unlike to the traditional CIP method,the derivative used in the presented scheme is obtained by using a high-order compact scheme originally proposed by Lele. The position of grid point along the characteristic curves can be obtained by using the semi-Lagrangian method.The evolution of the point value at the next time step is defined by using the obtained polynomial at the position of the grid point.To test the high accurate property of the scheme,we applied to solve the Burger's and KdV equations. A series of numerical experiments are given,and numerical results also verify the effectiveness of the new scheme.In the end,this scheme is also extended to solve one dimensional Euler equations.
In the second part,we discuss a characteristic-based finite volume method for hyperbolic conservation laws.We focus on the Euler system due to its importance in gas dynamics.In the scheme,the spatial derivatives are discritized by a finite volume method,while in time,the Simpson's quadrature rule is used.The point values at the cell boundaries are obtained by using the Central WENO reconstruction.The method is composed of the following steps:the position of the grid points along the characteristic curves are computed by using the third or fourth order Runge-Kutta method,while the polynomial is obtained by using the third order or fifth order CWENO reconstruction. Then,we obtain the grid point values by using the polynomial function at the point position.Finally,the numerical flux in the new finite volume scheme is obtained based on the point value.The new high resolution finite volume method is combined with the high resolution property of the characteristic method and the non-oscillatory property of the CWENO method.Some classical tests for both scalar and Euler conservations laws in one dimension are performed to verify he accuracy and convergence of the present scheme.
In the end,some benchmark tests of shallow wave equations are also adopted to verify the presented finite volume scheme,such as 1D dam-break with large depth difference,left critical rarefaction,two rarefactions and nearly dry bed,dam-break problem with dry bed and the generation of dry bed in the middle.Furthermore,we also extend the scheme to 2D conservation laws by using the dimensional splitting approach.For the 2D dam-break problems,the performance of the present scheme has been compared with the WAF scheme and other high-order schemes.
引文
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