A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem
A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem
复制标题
DOI：
10.1137/s003614290138471x
复制
发表时间：
2001-04
期刊：
影响因子：
--
通讯作者：
N. Kopteva;M. Stynes
中科院分区：
文献类型：
--
作者：
N. Kopteva;M. Stynes
考虑一类拟线性守恒对流扩散两点边值问题。数值求解时，采用迎风有限差分格式。所使用的网格具有固定数目（N+1）的节点，并且最初是均匀的，但是其节点使用基于当前计算的分段线性解的弧长的等分布的de Boor的简单算法自适应地移动。它是第一次证明，网格存在均匀分布弧长沿着的多边形的解决方案曲线，相应的计算解决方案是一阶精度，均匀在$\vareps $，其中$\vareps $是扩散系数。在边值问题是线性的情况下，如果N足够大，且与$\vareps $无关，则在算法的$O（{\rm ln}（1/\vareps）/{\rm ln} N）$迭代后，计算解的分段线性插值在$\vareps $中一致地在$L^\infty[0，1]$范数下达到一阶精度.数值实验支持我们的理论结果。
A quasi-linear conservative convection-diffusion two-point boundary value problem is considered. To solve it numerically, an upwind finite difference scheme is applied. The mesh used has a fixed number (N+1) of nodes and is initially uniform, but its nodes are moved adaptively using a simple algorithm of de Boor based on equidistribution of the arc-length of the current computed piecewise linear solution. It is proved for the first time that a mesh exists that equidistributes the arc-length along the polygonal solution curve and that the corresponding computed solution is first-order accurate, uniformly in $\varepsilon$, where $\varepsilon$ is the diffusion coefficient. In the case when the boundary value problem is linear, if N is sufficiently large independently of $\varepsilon$, it is shown that after $O({\rm ln}(1/\varepsilon)/{\rm ln} N)$ iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves first-order accuracy in the $L^\infty[0,1]$ norm uniformly in $\varepsilon$. Numerical experiments are presented that support our theoretical results.