The subject of this paper is the design of efficient and stable spectral methods for time-dependent partial differential equations in unit balls. We commence by sketching the desired features of a spectral method, which is defined by a choice of an orthonormal basis acting in the spatial domain. We continue by considering in detail the choice of a $W$-function basis in a disc in $\mathbb{R}^2$. This is a nontrivial issue because of a clash between two objectives: skew symmetry of the differentiation matrix (which ensures inter alia that the method is stable) and the correct behaviour at the origin. We resolve it by representing the underlying space as an affine space and splitting the underlying functions. This is generalised to any dimension $d \geq 2$ in a natural manner and the paper is concluded with numerical examples that demonstrate how our choice of basis attains the best outcome out of a number of alternatives.

翻译：本文研究针对单位球中时间依赖偏微分方程的高效且稳定的谱方法设计。我们首先概述谱方法应具备的理想特性，该方法由空间域中正交基函数的选择定义。随后详细探讨二维圆盘中W函数基的选择问题。由于两个目标之间存在冲突：即微分矩阵的反对称性（确保方法稳定性）与原点处正确行为的矛盾，使得该问题具有非平凡性。我们通过将底层空间表示为仿射空间并对函数进行分裂来解决这一冲突。该方法可自然推广至任意d≥2维度，最后通过数值算例证明，相较于多种备选方案，我们选择的基函数能够实现最优结果。