一阶双曲方程和对流占优扩散方程的耗散谱元法
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摘要
本文主要讨论一阶双曲方程的耗散谱元法和间断耗散谱元法,以及耗散谱元法在对流占优的扩散方程上的应用.
首先讨论一阶双曲方程的耗散谱方法(DSM)和耗散谱元法(DPSEM).
对于带周期边界条件和带Dirichlet边界条件的一阶线性变系数双曲方程,分别构造Fourier DSM和Legendre DSM,通过收敛性分析,得到了空间方向的拟最优估计.数值例子主要表明：相对于传统的谱方法,该方法在处理边界及其附近的奇性方面有一定的优势.
以此为基础,本文构造了二维问题的DPSEM,用Crank-Nicolson (CN)格式进行时间离散,并对半离散和全离散格式进行了收敛性分析.数值结果验证了该方法对于二维光滑问题具有谱精度,对于具有比较复杂的边界条件的问题同样有效,对于有限光滑问题比传统的谱元法(SEM)有更好的结果.
本文还将Legendre DPSEM应用于非线性周期双曲方程,并对系数进行Chebyshev-Gauss-Labatto插值处理,以减少计算量.分析了半离散和Crank-Nicolson/Leap-Frog (CN/LF)全离散格式的收敛性,并给出了数值例子.光滑算例的结果显示出DPSEM对于非线性问题同样具有谱精度；对于某些在区间内部具有有限光滑性和在边界处具有弱间断性的算例,当计算时间较长时,DPSEM能够得到比SEM更好的结果.
然后讨论一阶双曲方程的间断耗散谱元法(DDSEM).
对于线性方程,用DDSEM进行空间离散.用CN格式对半离散格式进行时间离散；分析了其稳定性和收敛性,还设计了一种算法,于通常所用的基中引入一组“间断”基,使得各单元的问题可以并行求解；数值例子显示出DDSEM结合了间断机制和耗散机制的优点.为了提高格式在时间方向的收敛速度,用间断Galerkin格式进行时间离散,也给出了稳定性和收敛性分析以及数值结果,并与CN-DDSEM进行了比较.之后简要讨论了时空间断耗散谱元法,进行了稳定性和收敛性分析,得到了时空方向的拟最优误差估计.
对于非线性双曲方程的DDSEM,时间方向用CN/LF格式进行离散.给出的数值算例将该格式与SEM和前面讨论的DPSEM进行了比较：一方面验证了DPSEM对于非周期问题的有效性,一方面显示出对于弱奇性解,DDSEM在流入边界所在单元内的误差比DPSEM,特别是比SEM小.
最后,本文将DPSEM应用于非线性对流扩散方程,并用Chebyshev-Gauss-Labatto插值对系数进行了处理.分析了γ≥γ0>0情况下二维问题的全离散格式的收敛性,得到了空间方向的拟最优估计.还给出了一维和二维的算例,显示出在扩散项系数γ比较小的情况下,即对于对流占优的扩散方程,DPSEM能够发挥其所带的耗散项的作用,在一定程度上控制误差的积累,得到比SEM更好的结果.
This work mainly discusses the dissipative spectral element method (DPSEM) and thediscontinuous dissipative spectral element method (DDSEM) for the first order hyperbolicequations, and the application of the DPSEM to the nonlinear advection-dominant difusionequation.
Firstly, the dissipative spectral method (DSM) and the DPSEM for the first order hyper-bolic equations are discussed.
We construct Fourier DSM and Legendre DSM for the first order linear variable coefcientshyperbolic equations with the periodic and the Dirichlet boundary conditions respectively, andobtain quasi-optimal error estimates in spacial direction for both schemes. Numerical examplesshow some superiorities of the underlying method over the standard spectral method in dealingwith the singularity nearby the boundaries.
Taking the DSM as the foundation, we construct the DPSEM for two-dimensional hyper-bolic equation. Convergencies are established for both the semi-discrete and the Crank-Nicolson(CN) fully discrete schemes. Numerical results verify the spectral accuracy of the scheme ap-plying to smooth solution, the validity in dealing with complex boundary conditions, and somesuperiority over the SEM in dealing with the solutions of limit smoothness.
Furthermore, we apply the Legendre DPSEM to the nonlinear periodic hyperbolic equa-tion, treating the coefcients with the Chebyshev-Gauss-Labatto collocation method to reducecomputational capacity. Convergencies are established for both the semi-discrete and the Crank-Nicolson/Leap-Frog (CN/LF) fully discrete schemes, and numerical examples are given. Thesmooth example shows that the DPSEM has spectral accuracy for nonlinear problem as well aslinear equations. An example which has limit smoothness in the interval and weak discontinuityat the boundaries shows that, for such a problem, the DPSEM can obtain better results thanthe SEM when taking long time calculation.
Secondly, the DDSEM for the first order hyperbolic equations is discussed.
To the linear equation, we apply the DDSEM in space discretization. First, the CNscheme is used in time discretization. The stability and the convergence of the CN-DDSEM areestablished. Furthermore, we devise an algorithm by introducing an additional ‘discontinuous’basis, so that the problem in each element can be solved in parallel. What can be seen fromour numerical results is that, the discontinuous mechanism and the dissipative mechanism can localize and homogenize the errors respectively, and the DDSEM summarizes both the virtues.Next, we use the discontiuous Galerkin (DG) scheme in time discretization to improve theaccuracy in temporal direction. Analysis of the stability and convergence of the DG-DDSEMand numerical results are given. In the examples, the CN-DDSEM and the DG-DDSEM arecompared. After that, we discuss the space-time DDSEM, of which the stability and the quasi-optimal error estimate in both spacial and temporal directions are obtained.
To the nonlinear equation, we apply the DDSEM with CN/LF time discretization. Anexample is given to compare the DDSEM with the SEM and the DPSEM discussed before: onone hand, it verifies the validity of the DPSEM for non-periodic problems; on the other hand,the results show that, for the solutions having some singularity, the error in the element withinflow boundary in of the DDSEM are smaller than that of the DPSEM, especially smaller thanthe SEM.
At last, the DPSEM is used to solve the nonlinear convection-difusion equation, in whichthe coefcients are treated with the Chebyshev-Gauss-Labatto collocation method. Quasi-optimal error estimate in spacial direction of the fully discrete scheme for the two-dimensionalequation in the case of γ≥γ0>0is obtained. Besides, we give out examples of both one dimen-sion and two dimensions. From the numerical results, we find that, when the coefcient γ of thedifusion term is small enough, the dissipative term, which can reduce the error accumulationto a certain extent, makes the DPSEM perform better than the SEM.
首先讨论一阶双曲方程的耗散谱方法(DSM)和耗散谱元法(DPSEM).
对于带周期边界条件和带Dirichlet边界条件的一阶线性变系数双曲方程,分别构造Fourier DSM和Legendre DSM,通过收敛性分析,得到了空间方向的拟最优估计.数值例子主要表明：相对于传统的谱方法,该方法在处理边界及其附近的奇性方面有一定的优势.
以此为基础,本文构造了二维问题的DPSEM,用Crank-Nicolson (CN)格式进行时间离散,并对半离散和全离散格式进行了收敛性分析.数值结果验证了该方法对于二维光滑问题具有谱精度,对于具有比较复杂的边界条件的问题同样有效,对于有限光滑问题比传统的谱元法(SEM)有更好的结果.
本文还将Legendre DPSEM应用于非线性周期双曲方程,并对系数进行Chebyshev-Gauss-Labatto插值处理,以减少计算量.分析了半离散和Crank-Nicolson/Leap-Frog (CN/LF)全离散格式的收敛性,并给出了数值例子.光滑算例的结果显示出DPSEM对于非线性问题同样具有谱精度；对于某些在区间内部具有有限光滑性和在边界处具有弱间断性的算例,当计算时间较长时,DPSEM能够得到比SEM更好的结果.
然后讨论一阶双曲方程的间断耗散谱元法(DDSEM).
对于线性方程,用DDSEM进行空间离散.用CN格式对半离散格式进行时间离散；分析了其稳定性和收敛性,还设计了一种算法,于通常所用的基中引入一组“间断”基,使得各单元的问题可以并行求解；数值例子显示出DDSEM结合了间断机制和耗散机制的优点.为了提高格式在时间方向的收敛速度,用间断Galerkin格式进行时间离散,也给出了稳定性和收敛性分析以及数值结果,并与CN-DDSEM进行了比较.之后简要讨论了时空间断耗散谱元法,进行了稳定性和收敛性分析,得到了时空方向的拟最优误差估计.
对于非线性双曲方程的DDSEM,时间方向用CN/LF格式进行离散.给出的数值算例将该格式与SEM和前面讨论的DPSEM进行了比较：一方面验证了DPSEM对于非周期问题的有效性,一方面显示出对于弱奇性解,DDSEM在流入边界所在单元内的误差比DPSEM,特别是比SEM小.
最后,本文将DPSEM应用于非线性对流扩散方程,并用Chebyshev-Gauss-Labatto插值对系数进行了处理.分析了γ≥γ0>0情况下二维问题的全离散格式的收敛性,得到了空间方向的拟最优估计.还给出了一维和二维的算例,显示出在扩散项系数γ比较小的情况下,即对于对流占优的扩散方程,DPSEM能够发挥其所带的耗散项的作用,在一定程度上控制误差的积累,得到比SEM更好的结果.
This work mainly discusses the dissipative spectral element method (DPSEM) and thediscontinuous dissipative spectral element method (DDSEM) for the first order hyperbolicequations, and the application of the DPSEM to the nonlinear advection-dominant difusionequation.
Firstly, the dissipative spectral method (DSM) and the DPSEM for the first order hyper-bolic equations are discussed.
We construct Fourier DSM and Legendre DSM for the first order linear variable coefcientshyperbolic equations with the periodic and the Dirichlet boundary conditions respectively, andobtain quasi-optimal error estimates in spacial direction for both schemes. Numerical examplesshow some superiorities of the underlying method over the standard spectral method in dealingwith the singularity nearby the boundaries.
Taking the DSM as the foundation, we construct the DPSEM for two-dimensional hyper-bolic equation. Convergencies are established for both the semi-discrete and the Crank-Nicolson(CN) fully discrete schemes. Numerical results verify the spectral accuracy of the scheme ap-plying to smooth solution, the validity in dealing with complex boundary conditions, and somesuperiority over the SEM in dealing with the solutions of limit smoothness.
Furthermore, we apply the Legendre DPSEM to the nonlinear periodic hyperbolic equa-tion, treating the coefcients with the Chebyshev-Gauss-Labatto collocation method to reducecomputational capacity. Convergencies are established for both the semi-discrete and the Crank-Nicolson/Leap-Frog (CN/LF) fully discrete schemes, and numerical examples are given. Thesmooth example shows that the DPSEM has spectral accuracy for nonlinear problem as well aslinear equations. An example which has limit smoothness in the interval and weak discontinuityat the boundaries shows that, for such a problem, the DPSEM can obtain better results thanthe SEM when taking long time calculation.
Secondly, the DDSEM for the first order hyperbolic equations is discussed.
To the linear equation, we apply the DDSEM in space discretization. First, the CNscheme is used in time discretization. The stability and the convergence of the CN-DDSEM areestablished. Furthermore, we devise an algorithm by introducing an additional ‘discontinuous’basis, so that the problem in each element can be solved in parallel. What can be seen fromour numerical results is that, the discontinuous mechanism and the dissipative mechanism can localize and homogenize the errors respectively, and the DDSEM summarizes both the virtues.Next, we use the discontiuous Galerkin (DG) scheme in time discretization to improve theaccuracy in temporal direction. Analysis of the stability and convergence of the DG-DDSEMand numerical results are given. In the examples, the CN-DDSEM and the DG-DDSEM arecompared. After that, we discuss the space-time DDSEM, of which the stability and the quasi-optimal error estimate in both spacial and temporal directions are obtained.
To the nonlinear equation, we apply the DDSEM with CN/LF time discretization. Anexample is given to compare the DDSEM with the SEM and the DPSEM discussed before: onone hand, it verifies the validity of the DPSEM for non-periodic problems; on the other hand,the results show that, for the solutions having some singularity, the error in the element withinflow boundary in of the DDSEM are smaller than that of the DPSEM, especially smaller thanthe SEM.
At last, the DPSEM is used to solve the nonlinear convection-difusion equation, in whichthe coefcients are treated with the Chebyshev-Gauss-Labatto collocation method. Quasi-optimal error estimate in spacial direction of the fully discrete scheme for the two-dimensionalequation in the case of γ≥γ0>0is obtained. Besides, we give out examples of both one dimen-sion and two dimensions. From the numerical results, we find that, when the coefcient γ of thedifusion term is small enough, the dissipative term, which can reduce the error accumulationto a certain extent, makes the DPSEM perform better than the SEM.
引文
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