arXiv 2302.042172023-02-08math.NAcs.NA

不定常谱方法的正交系

Orthogonal systems for time-dependent spectral methods

Arieh Iserles

摘要：

本文研究实区间上Dirichlet边界条件为零的标准正交方程组。更具体地说，我们的兴趣是在系统的斜对称微分矩阵（这不包括正交多项式）。我们认为这样的系统的一个简单的建设，并追求其后果。一般来说，给定任何$\mathrm{C}^1(a,b)$权函数，使得$w(a)=w(b)=0$，我们可以生成具有反对称微分矩阵的标准正交系统。除了情况$a=-\infty$，$b=+\infty$，只有有限数量的权力，该矩阵是有界的，我们建立了一个连接之间的属性的权重函数和有界性。特别是，我们详细研究了两个权重函数：拉盖尔权函数$x^α\mathrm{e}^{-x}$的$x>0$和$α>0$和超球权函数$(1-x^2)^α$，$x\in(-1,1)$，$α>0$，并建立其属性。这两个权重都有一个最受欢迎的特性，即{ \em可分性，\/}，它允许快速计算。近似的质量是高度敏感的$α$的选择，我们讨论如何选择最佳的这个参数，根据零边界条件的数量。

英文摘要：

This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any $\mathrm{C}^1(a,b)$ weight function such that $w(a)=w(b)=0$, we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case $a=-\infty$, $b=+\infty$, only a limited number of powers of that matrix is bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function $x^α\mathrm{e}^{-x}$ for $x>0$ and $α>0$ and the ultraspherical weight function $(1-x^2)^α$, $x\in(-1,1)$, $α>0$, and establish their properties. Both weights share a most welcome feature of {\em separability,\/} which allows for fast computation. The quality of approximation is highly sensitive to the choice of $α$ and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.

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英文摘要：

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