| Abstract | 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$~\theta~$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$~\theta$~趋近于~$1/2$~时, 格式的稳定性条件会明显改善.
而当~$\theta$~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. |
| Other Abstract | 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-implicit
$\theta$-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 $\theta$
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... |
