基于多种指示子的杂交WENO格式
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摘要
求解非线性双曲守恒律方程和对流占优问题的WENO (weighted essentially non-oscillatory)格式具有精度高,同时保持稳定的、本质无振荡的和陡峭的间断过渡等优点,特别适合含有强激波和复杂光滑解结构的问题。WENO格式是鲁棒的,并且不需要人为参数。这些优点使得WENO格式迅速发展成为计算流体力学的主流方法之一,并已成功应用到下列领域：计算天文学、半导体器件模拟、计算生物学、交通流模型。有限差分WENO格式的主要思想是将各个候选模板上的低阶数值流通量逼近加以组合以得到高阶逼近。分配给每个候选模板的权(网格点值的非线性函数),是有限差分WENO格式成功的关键所在。对于方程组,WENO格式须借助局部特征分解步骤和流通量分裂以避免数值振荡。但是非线性权计算和局部特征分解步骤的成本是很高的,并且随着方程个数和空间维数的增加,计算成本增加趋势更加明显。这使得研究高效WENO格式显得尤为必要,而且目前这方面的研究还相对较少,所以本文选取它作为研究内容。
本文主要研究杂交WENO格式,将WENO重构(使用非线性权)与简单的迎风线性重构(使用线性权)相结合,其主要思想是在间断区域使用WENO重构捕捉间断,而在光滑区域使用高效的迎风线性重构,以避免使用非线性权和局部特征分解步骤。从而达到在保持WENO格式原有良好性质的同时,节省计算成本的目的。由于WENO重构与线性迎风重构关系密切,所以与其它杂交WENO格式相比较,本文中的杂交WENO格式在思想上更加直接,并且两种重构的精度匹配更加一致,所以数值流通量更加光滑。杂交WENO格式的一个重要组成部分,是能够自动、准确捕捉“坏单元”(包含间断的单元)的指示子。在本文中我们主要借鉴DG (discontinuous Galerkin)有限元方法中的限制器,来构造适用于有限差分方法的坏单元指示子,通过大量的数值试验,为杂交WENO格式设计和挑选出实用的坏单元指示子,使得基于这些指示子的杂交WENO格式在保持原WENO格式的良好性质的同时,还比原WENO格式具有更高的计算效率,使得杂交WENO格式更具有实际应用价值。
我们首先针对一维可压缩Euler方程,利用经典的数值算例测试了基于九种不同坏单元指示子的杂交WENO格式,挑选出四种表现较好的指示子：ATV、TVB、MR和KXRCF,因为这些指示子在CPU时间、数值解误差以及数值流通量逼近过程中WENO重构使用百分比方面都优于其它指示子。
随后,我们将基于这四种指示子的杂交WENO格式推广到Burgers方程、多维Euler方程,数值试验结果令人满意。
针对带有源项的双曲守恒律方程：浅水波方程和浅水中污染物输运方程,我们设计了基于指示子的杂交well-balanced WENO格式。理论分析和数值试验皆表明杂交well-balanced WENO格式能够保持精确守恒属性,该属性对于流通量梯度与源项之间的平衡显得至关重要。
为了更好地处理一般物理区域,我们首先对物理区域进行曲线网格剖分,然后借助坐标变换,将一般物理区域映射为笛卡尔计算区域,同时将物理区域中的曲线网格映射为计算区域中的一致网格。接下来把物理空间中的基本方程变换到计算空间中,最后我们把杂交WENO格式推广到计算空间中的基本方程上。数值结果十分理想。
大量的多维问题、不同类型问题以及基于不同网格的数值算例,表明基于ATV、TVB、MR以及KXRCF指示子的杂交WENO格式在保持WENO格式原有良好属性的同时,能够较可观地节省计算成本,提高计算效率。
总之,基于ATV、TVB、MR以及KXRCF指示子的杂交WENO格式是高效的、鲁棒的。
The weighted essentially non-oscillatory (WENO) schemes for solving nonlinear hyperbolic conservation laws and convection dominated problems are high-order accu-rate and maintain stable, essentially non-oscillatory and sharp discontinuity transition. The schemes are thus especially suitable for problems containing both strong disconti-nuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. With the above properties, WENO schemes become the key numerical method in the computational fluid dynamics domain, and have been generalized to the following applications in areas including computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology. A key idea in finite difference WENO schemes is a combination of lower order fluxes on candidate stencil to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is cru-cial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious numerical oscillations. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. With the increase of the number of equation and space dimension, the trend of the increase of computational cost is more evident. This draw-back makes it necessary to implement the research for the efficient WENO schemes. At the same time, there have been little efforts in the literature to overcome the drawback, so we carry out the work in this dissertation.
We focus our research on the hybrid WENO schemes to hybridize the WENO reconstruction (using nonlinear weights) and the simple up-wind linear reconstruction (using linear weights). The key idea is to use the WENO reconstruction in the dis-continuous regions to capture discontinuities, and to use the efficient up-wind linear reconstruction in order to avoid the usage of nonlinear weight and the local character-istic decomposition procedure. Our aim is to apply the hybrid WENO schemes to save computational cost considerably and at the same time to maintain the original good properties of the WENO scheme. Compared with the other hybrid WENO schemes, the hybrid WENO scheme of this paper has the following properties:its key idea is more direct than others, due to the close relation between the WENO reconstruction and the up-wind linear reconstruction; the accuracy of the two type reconstructions is consistent, so the numerical fluxes are smoother. An important component of the hy-brid scheme is a "troubled-cell" (cell contains discontinuity) indicator to automatically identify where the discontinuity of the solution is. In this paper, we reconstruct the troubled-cell indicators mainly based on the limiters from the discontinuous Galerkin (DG) methods. By extensive numerical experiments, we design and choose practical troubled-cell indicators for the hybrid WENO schemes, such that the hybrid WENO schemes with the indicators have higher efficiency than the original WENO schemes and maintain the good properties of the original WENO schemes at the same time and have practical value.
Firstly, for the one-dimensional compressible Euler equations, we test the hybrid WENO schemes with none indicators by benchmark numerical examples. We find that ATV, TVB, MR and KXRCF indicators are better than others. For they result in little CPU time, more accurate numerical solutions and smaller percentages of fluxes by WENO reconstruction than the remaining indicators.
Subsequently, we extend the hybrid WENO schemes with the above four indi-cators to Burgers equation, multi-dimensional Euler equations. The numerical results indicate that the hybrid WENO schemes can save computational cost considerabley.
For the hyperbolic conservation law equation with source terms:shallow water equations and pollutant transport equations in shallow water, we design hybrid well-balanced WENO schemes with indicators. Theoretical analysis and numerical test ver-ify that the hybrid well-balanced WENO schemes can maintain the exact conservation property (exact C-property). The property is important for the balance between the flux gradient and the source terms.
In order to handle the general physical domain well, we decompose the physi- cal domain into curvilinear grids. A given coordinate transformation maps the general physical domain to a Cartesian computational domain and maps the curvilinear grid in physical domain to a regular grid in computational domain. Meanwhile, we trans-form the governing equation in physical space to computational space. At last, we generalize the hybrid WENO schemes to the hyperbolic conservation law equation in computational space. The numerical results are desirable.
Extensive numerical examples based on multi-dimensional, on different type prob-lems and on different type grids suggest that the hybrid WENO schemes with indica-tors can save computational cost considerably and maintain the property of the original WENO schemes at the same time.
In summary, the hybrid WENO schemes with the ATV, TVB, MR and KXRCF indicators is efficiency effective and robust.
本文主要研究杂交WENO格式,将WENO重构(使用非线性权)与简单的迎风线性重构(使用线性权)相结合,其主要思想是在间断区域使用WENO重构捕捉间断,而在光滑区域使用高效的迎风线性重构,以避免使用非线性权和局部特征分解步骤。从而达到在保持WENO格式原有良好性质的同时,节省计算成本的目的。由于WENO重构与线性迎风重构关系密切,所以与其它杂交WENO格式相比较,本文中的杂交WENO格式在思想上更加直接,并且两种重构的精度匹配更加一致,所以数值流通量更加光滑。杂交WENO格式的一个重要组成部分,是能够自动、准确捕捉“坏单元”(包含间断的单元)的指示子。在本文中我们主要借鉴DG (discontinuous Galerkin)有限元方法中的限制器,来构造适用于有限差分方法的坏单元指示子,通过大量的数值试验,为杂交WENO格式设计和挑选出实用的坏单元指示子,使得基于这些指示子的杂交WENO格式在保持原WENO格式的良好性质的同时,还比原WENO格式具有更高的计算效率,使得杂交WENO格式更具有实际应用价值。
我们首先针对一维可压缩Euler方程,利用经典的数值算例测试了基于九种不同坏单元指示子的杂交WENO格式,挑选出四种表现较好的指示子：ATV、TVB、MR和KXRCF,因为这些指示子在CPU时间、数值解误差以及数值流通量逼近过程中WENO重构使用百分比方面都优于其它指示子。
随后,我们将基于这四种指示子的杂交WENO格式推广到Burgers方程、多维Euler方程,数值试验结果令人满意。
针对带有源项的双曲守恒律方程：浅水波方程和浅水中污染物输运方程,我们设计了基于指示子的杂交well-balanced WENO格式。理论分析和数值试验皆表明杂交well-balanced WENO格式能够保持精确守恒属性,该属性对于流通量梯度与源项之间的平衡显得至关重要。
为了更好地处理一般物理区域,我们首先对物理区域进行曲线网格剖分,然后借助坐标变换,将一般物理区域映射为笛卡尔计算区域,同时将物理区域中的曲线网格映射为计算区域中的一致网格。接下来把物理空间中的基本方程变换到计算空间中,最后我们把杂交WENO格式推广到计算空间中的基本方程上。数值结果十分理想。
大量的多维问题、不同类型问题以及基于不同网格的数值算例,表明基于ATV、TVB、MR以及KXRCF指示子的杂交WENO格式在保持WENO格式原有良好属性的同时,能够较可观地节省计算成本,提高计算效率。
总之,基于ATV、TVB、MR以及KXRCF指示子的杂交WENO格式是高效的、鲁棒的。
The weighted essentially non-oscillatory (WENO) schemes for solving nonlinear hyperbolic conservation laws and convection dominated problems are high-order accu-rate and maintain stable, essentially non-oscillatory and sharp discontinuity transition. The schemes are thus especially suitable for problems containing both strong disconti-nuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. With the above properties, WENO schemes become the key numerical method in the computational fluid dynamics domain, and have been generalized to the following applications in areas including computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology. A key idea in finite difference WENO schemes is a combination of lower order fluxes on candidate stencil to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is cru-cial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious numerical oscillations. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. With the increase of the number of equation and space dimension, the trend of the increase of computational cost is more evident. This draw-back makes it necessary to implement the research for the efficient WENO schemes. At the same time, there have been little efforts in the literature to overcome the drawback, so we carry out the work in this dissertation.
We focus our research on the hybrid WENO schemes to hybridize the WENO reconstruction (using nonlinear weights) and the simple up-wind linear reconstruction (using linear weights). The key idea is to use the WENO reconstruction in the dis-continuous regions to capture discontinuities, and to use the efficient up-wind linear reconstruction in order to avoid the usage of nonlinear weight and the local character-istic decomposition procedure. Our aim is to apply the hybrid WENO schemes to save computational cost considerably and at the same time to maintain the original good properties of the WENO scheme. Compared with the other hybrid WENO schemes, the hybrid WENO scheme of this paper has the following properties:its key idea is more direct than others, due to the close relation between the WENO reconstruction and the up-wind linear reconstruction; the accuracy of the two type reconstructions is consistent, so the numerical fluxes are smoother. An important component of the hy-brid scheme is a "troubled-cell" (cell contains discontinuity) indicator to automatically identify where the discontinuity of the solution is. In this paper, we reconstruct the troubled-cell indicators mainly based on the limiters from the discontinuous Galerkin (DG) methods. By extensive numerical experiments, we design and choose practical troubled-cell indicators for the hybrid WENO schemes, such that the hybrid WENO schemes with the indicators have higher efficiency than the original WENO schemes and maintain the good properties of the original WENO schemes at the same time and have practical value.
Firstly, for the one-dimensional compressible Euler equations, we test the hybrid WENO schemes with none indicators by benchmark numerical examples. We find that ATV, TVB, MR and KXRCF indicators are better than others. For they result in little CPU time, more accurate numerical solutions and smaller percentages of fluxes by WENO reconstruction than the remaining indicators.
Subsequently, we extend the hybrid WENO schemes with the above four indi-cators to Burgers equation, multi-dimensional Euler equations. The numerical results indicate that the hybrid WENO schemes can save computational cost considerabley.
For the hyperbolic conservation law equation with source terms:shallow water equations and pollutant transport equations in shallow water, we design hybrid well-balanced WENO schemes with indicators. Theoretical analysis and numerical test ver-ify that the hybrid well-balanced WENO schemes can maintain the exact conservation property (exact C-property). The property is important for the balance between the flux gradient and the source terms.
In order to handle the general physical domain well, we decompose the physi- cal domain into curvilinear grids. A given coordinate transformation maps the general physical domain to a Cartesian computational domain and maps the curvilinear grid in physical domain to a regular grid in computational domain. Meanwhile, we trans-form the governing equation in physical space to computational space. At last, we generalize the hybrid WENO schemes to the hyperbolic conservation law equation in computational space. The numerical results are desirable.
Extensive numerical examples based on multi-dimensional, on different type prob-lems and on different type grids suggest that the hybrid WENO schemes with indica-tors can save computational cost considerably and maintain the property of the original WENO schemes at the same time.
In summary, the hybrid WENO schemes with the ATV, TVB, MR and KXRCF indicators is efficiency effective and robust.
引文
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