论文标题
第二个衍生物的逐件概述：奇异运算符的伪内，并重新访问第六阶准确的窄模式操作员
Summation-by-parts approximations of the second derivative: Pseudoinverses of singular operators and revisiting the sixth order accurate narrow-stencil operator
论文作者
论文摘要
我们考虑了第二个衍生物的有限差近似值，在泊松方程，热方程和波方程中举例说明。有限的差异操作员满足了按零件的总结属性，该属性模仿了逐个部分的集成。由于操作员近似第二个导数，因此它们是构造的单数。为了强加边界条件，使用同时近似项修改这些操作员。对于大多数边界条件选择，这使得修改的矩阵非singular。最近，得出了此类矩阵的反面。但是，当考虑两个边界上的neumann边界条件时，修改的矩阵仍然是单数。对于此类矩阵，我们得出了摩尔 - 佩罗斯伪源的明确表达，可用于解决椭圆问题和一些时间依赖性问题。这种新的伪内词有效的条件是，修改后的矩阵的特征值不超过一个零。我们已经用免费参数重建了第六阶精确的窄模式操作员，并证明可能发生一个以上的零特征值。我们已经对自由参数进行了详细的分析，以改善第二个衍生算子的属性。我们通过数值实验补充了派生，以证明新的第二个衍生算子的改进。
We consider finite difference approximations of the second derivative, exemplified in Poisson's equation, the heat equation and the wave equation. The finite difference operators satisfy a summation-by-parts property, which mimics the integration-by-parts. Since the operators approximate the second derivative, they are singular by construction. To impose boundary conditions, these operators are modified using Simultaneous Approximation Terms. This makes the modified matrices non-singular, for most choices of boundary conditions. Recently, inverses of such matrices were derived. However, when considering Neumann boundary conditions on both boundaries, the modified matrix is still singular. For such matrices, we have derived an explicit expression for the Moore-Penrose pseudoinverse, which can be used for solving elliptic problems and some time-dependent problems. The condition for this new pseudoinverse to be valid, is that the modified matrix does not have more than one zero eigenvalue. We have reconstructed the sixth order accurate narrow-stencil operator with a free parameter and show that more than one zero eigenvalue can occur. We have performed a detailed analysis on the free parameter to improve the properties of the second derivative operator. We complement the derivations by numerical experiments to demonstrate the improvements of the new second derivative operator.
