偏微分方程数值计算及增量未知元方法研究
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摘要
本文基于微分方程的有限差分技术以及一致网格增量未知元方法,分别对一维和二维具有时间依赖系数的热方程以及一类一般的三维对流扩散方程进行了不同的研究。由于一致网格增量未知元方法可以很好地降低矩阵条件数,所以该方法的优越性在我们的理论分析和数值实验中都很好地体现了出来。非一致网格作为一种更为灵活的形式,对于许多问题,特别是边界层问题的求解,有着一致网格所无法比拟的优势。相应地非一致网格上的增量未知元方法便自然地引起了我们的注意。所以在本文的后半部分,我们从理论方面,以Dirichlet问题为例,对一维和二维非一致网格增量未知元方法下的系数矩阵条件数进行了详细得分析,并用数值算例对我们的理论分析进行了验证。
对于具有时间依赖系数的一维热方程,我们提出了一类增量未知元型半隐θ格式,仔细分析了这一新格式的稳定性,并对其误差进行了估计。结果显示,当θ趋近于1/2时,格式的稳定性条件会明显改善.而当θ趋近于零时,我们得到了一个能帮助我们恢复初始格式误差的条件。对于二维情形,我们依然用有限差分进行离散,构造了一类交替方向增量未知元型半隐(ADIUSI)格式。并在傅立叶方法的帮助下,对格式的稳定性进行了详细的分析。数值试验验证了理论分析的正确性,其结果表明这一新的格式,在某些问题上,会比经典的交替方向格式更为有效。
对于一类一般形式的三维对流扩散方程,在有限差分和增量未知元方法下,可以得到一个增量未知元型正定但非对称的线性方程组。其系数矩阵条件数要远远优于不用增量未知元方法的情形。考虑到该方法的这一优点,我们在文中将其与几种经典的迭代方法相结合,来求解上述线性系统.并从理论上对该系统的增量未知元型系数矩阵条件数进行了估计,然后通过数值试验验证了这几种增量未知元型迭代方法的有效性。
我们注意到上述的差分离散和增量未知元方法都是在一致网格上进行的,然而对于许多问题,例如边界层问题、流体力学问题等,一致网格上的差分离散已不能满足它们求解精度的需要。很自然地,要考虑非一致网格上的差分离散以及相应地非一致网格增量未知元方法。随之而来的问题是,这种非一致网格增量未知元方法能否也像一致网格增量未知元方法那样,能有效降低系数矩阵条件数?我们在本文后半部分对于该方法的一维和二维情形都进行了详细分析。理论结果表明该方法依然可以很有效地降低矩阵条件数,数值试验结果与我们的理论分析完全吻合。
In this paper, based on the finite difference discretization of partial differential equations and the advantage that the incremental unknowns(IU) on uniform meshes can reducing the condition number of coefficient matrix effectively, we study the heat equations with the time-dependent coefficients in the 1 and 2 dimensions and a class of generalized three dimensional convection-diffusion equations with this method. The effectiveness of this method are established by the theoretical analysis and the numerical results. But for many problems, especially the boundary layer problems, nonuniform meshes are more flexible and efficient than the uniform meshes. So, the discretization technique and then the incremental unknowns on nonuniform meshes(NIU) become more and more important. With the Dirichlet problem, we theoretically and numerically analyze the condition number of the coefficient matrix with NIU in dimension 1 and 2.
For the one dimensional heat equation with time-dependent coefficient, we propose a kind of IU-type semi-implicitθ-schemes and carefully study the stability, error estimation and condition number of these schemes. The theoretical analysis shows that a better stability condition was obtained when 9 close to 1/2. For the two dimensional case, we construct an alternating direction IU-type semi-implicit scheme. The stability condition of this new scheme is obtained with the Fourier method. Numerical results show that this new scheme is more efficient than the classical alternating directional scheme for some problems when r satisfies the stability condition.
With the finite difference discretization techniques and the IU method, we get a nonsymmetric and positive-definite linear system when considering a class of generalized three dimensional convection-diffusion equations. Considering that the condition number of this coefficient matrix is much better than the matrix without IU, we use this method in conjunction with several classical iterative methods to approximate the solution of the system. After estimating the condition number of IU-type coefficient matrix, we numerically confirm that these IU-type iterative methods are much more efficient.
Note that the finite difference discretization techniques and the IU method are defined on the uniform meshes. But for many problems, for example, the boundary layer or hydromechanics problems, the methods defined on the uniform mesh are no longer work very well. Hence the NIU method becomes more and more important. With the question that does the NIU method also can reduce the condition number of coefficient matrix as the IU method, we give a carefully analysis both in 1 and 2 dimensions. The theoretical results show that NIU method also work effectively, which was established by the numerical results.
对于具有时间依赖系数的一维热方程,我们提出了一类增量未知元型半隐θ格式,仔细分析了这一新格式的稳定性,并对其误差进行了估计。结果显示,当θ趋近于1/2时,格式的稳定性条件会明显改善.而当θ趋近于零时,我们得到了一个能帮助我们恢复初始格式误差的条件。对于二维情形,我们依然用有限差分进行离散,构造了一类交替方向增量未知元型半隐(ADIUSI)格式。并在傅立叶方法的帮助下,对格式的稳定性进行了详细的分析。数值试验验证了理论分析的正确性,其结果表明这一新的格式,在某些问题上,会比经典的交替方向格式更为有效。
对于一类一般形式的三维对流扩散方程,在有限差分和增量未知元方法下,可以得到一个增量未知元型正定但非对称的线性方程组。其系数矩阵条件数要远远优于不用增量未知元方法的情形。考虑到该方法的这一优点,我们在文中将其与几种经典的迭代方法相结合,来求解上述线性系统.并从理论上对该系统的增量未知元型系数矩阵条件数进行了估计,然后通过数值试验验证了这几种增量未知元型迭代方法的有效性。
我们注意到上述的差分离散和增量未知元方法都是在一致网格上进行的,然而对于许多问题,例如边界层问题、流体力学问题等,一致网格上的差分离散已不能满足它们求解精度的需要。很自然地,要考虑非一致网格上的差分离散以及相应地非一致网格增量未知元方法。随之而来的问题是,这种非一致网格增量未知元方法能否也像一致网格增量未知元方法那样,能有效降低系数矩阵条件数?我们在本文后半部分对于该方法的一维和二维情形都进行了详细分析。理论结果表明该方法依然可以很有效地降低矩阵条件数,数值试验结果与我们的理论分析完全吻合。
In this paper, based on the finite difference discretization of partial differential equations and the advantage that the incremental unknowns(IU) on uniform meshes can reducing the condition number of coefficient matrix effectively, we study the heat equations with the time-dependent coefficients in the 1 and 2 dimensions and a class of generalized three dimensional convection-diffusion equations with this method. The effectiveness of this method are established by the theoretical analysis and the numerical results. But for many problems, especially the boundary layer problems, nonuniform meshes are more flexible and efficient than the uniform meshes. So, the discretization technique and then the incremental unknowns on nonuniform meshes(NIU) become more and more important. With the Dirichlet problem, we theoretically and numerically analyze the condition number of the coefficient matrix with NIU in dimension 1 and 2.
For the one dimensional heat equation with time-dependent coefficient, we propose a kind of IU-type semi-implicitθ-schemes and carefully study the stability, error estimation and condition number of these schemes. The theoretical analysis shows that a better stability condition was obtained when 9 close to 1/2. For the two dimensional case, we construct an alternating direction IU-type semi-implicit scheme. The stability condition of this new scheme is obtained with the Fourier method. Numerical results show that this new scheme is more efficient than the classical alternating directional scheme for some problems when r satisfies the stability condition.
With the finite difference discretization techniques and the IU method, we get a nonsymmetric and positive-definite linear system when considering a class of generalized three dimensional convection-diffusion equations. Considering that the condition number of this coefficient matrix is much better than the matrix without IU, we use this method in conjunction with several classical iterative methods to approximate the solution of the system. After estimating the condition number of IU-type coefficient matrix, we numerically confirm that these IU-type iterative methods are much more efficient.
Note that the finite difference discretization techniques and the IU method are defined on the uniform meshes. But for many problems, for example, the boundary layer or hydromechanics problems, the methods defined on the uniform mesh are no longer work very well. Hence the NIU method becomes more and more important. With the question that does the NIU method also can reduce the condition number of coefficient matrix as the IU method, we give a carefully analysis both in 1 and 2 dimensions. The theoretical results show that NIU method also work effectively, which was established by the numerical results.
引文
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