A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law
A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law
复制标题
满足全局守恒定律的散度和梯度高阶近似的矩阵分析方法
DOI：
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发表时间：
2003
影响因子：
1.5
通讯作者：
R. Grone
中科院分区：
文献类型：
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作者：
José Castillo;R. Grone
一维，二阶有限差分近似的衍生物构造满足全球守恒律。在远离边界的地方创建二阶近似很简单，但是在边界附近获得适当的行为是困难的，即使是在均匀网格上的一维中。在这篇文章中，我们展示的技术，允许离散版本的发散和梯度算子，在边界上有高阶近似的建设。我们构造这样的离散化在一维的情况下，有四阶逼近的边界和内部。本文中的高阶模拟格式在边界点处的精度尽可能高（相对于带宽参数）。这保证了整体的高精度。此外，所描述的用于计算近似值的方法使用矩阵分析来简化各种模拟条件。这有助于使以前的办法更加明确。
这是为高维非均匀网格建立高阶近似的发散度和梯度的关键的第一步。
One-dimensional, second-order finite-difference approximations of the derivative are constructed which satisfy a global conservation law. Creating a second-order approximation away from the boundary is simple, but obtaining appropriate behavior near the boundary is difficult, even in one dimension on a uniform grid. In this article we exhibit techniques that allow the construction of discrete versions of the divergence and gradient operator that have high-order approximations at the boundary. We construct such discretizations in the one-dimensional situation which have fourth-order approximation both on the boundary and in the interior. The precision of the high-order mimetic schemes in this article is as high as possible at the boundary points (with respect to the bandwidth parameter). This guarantees an overall high order of accuracy. Furthermore, the method described for the calculation of the approximations uses matrix analysis to streamline the various mimetic conditions. This contributes to a marked clarity with respect to earlier approaches.
This is a crucial preliminary step in creating higher-order approximations of the divergence and gradient for nonuniform grids in higher dimensions.