论文标题
SBP-SAT有限差算子的逆，近似于第一个和第二个衍生物
Inverses of SBP-SAT finite difference operators approximating the first and second derivative
论文作者
论文摘要
考虑标量，一维对流方程和热方程。这些方程在空间中被离散，使用有限差方法满足逐个组合（SBP）属性。为了强加边界条件，我们使用一种称为同时近似项（SAT）的惩罚方法。总之，这产生了两个半分化方案，其中离散化矩阵分别近似于第一个和第二个衍生算子。离散矩阵取决于SAT处理中的自由参数。我们得出离散矩阵的倒置，将其解释为离散的绿色功能。以这种直接的方式，我们还准确地找出了使离散化矩阵单数的SAT参数的选择。在第二个导数情况下，结果表明，如果选择了惩罚参数，以使半分化方案是双重一致的，那么即使该方案是能量稳定的，离散化矩阵也会变得单数。与任意准确性顺序的SBP-SAT运算符的逆公式保持。对于第二阶和第四阶精确操作员，将明确提供倒置。
The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method satisfying summation-by-parts (SBP) properties. To impose the boundary conditions, we use a penalty method called simultaneous approximation term (SAT). Together, this gives rise to two semi-discrete schemes where the discretization matrices approximate the first and the second derivative operators, respectively. The discretization matrices depend on free parameters from the SAT treatment. We derive the inverses of the discretization matrices, interpreting them as discrete Green's functions. In this direct way, we also find out precisely which choices of SAT parameters that make the discretization matrices singular. In the second derivative case, it is shown that if the penalty parameters are chosen such that the semi-discrete scheme is dual consistent, the discretization matrix can become singular even when the scheme is energy stable. The inverse formulas hold for SBP-SAT operators of arbitrary order of accuracy. For second and fourth order accurate operators, the inverses are provided explicitly.
