We propose a Bernoulli-barycentric rational matrix collocation method for two-dimensional evolutionary partial differential equations (PDEs) with variable coefficients that combines Bernoulli polynomials with barycentric rational interpolations in time and space, respectively. The theoretical accuracy $O\left((2\pi)^{-N}+h_x^{d_x-1}+h_y^{d_y-1}\right)$ of our numerical scheme is proven, where $N$ is the number of basis functions in time, $h_x$ and $h_y$ are the grid sizes in the $x$, $y$-directions, respectively, and $0\leq d_x\leq \frac{b-a}{h_x},~0\leq d_y\leq\frac{d-c}{h_y}$. For the efficient solution of the relevant linear system arising from the discretizations, we introduce a class of dimension expanded preconditioners that take the advantage of structural properties of the coefficient matrices, and we present a theoretical analysis of eigenvalue distributions of the preconditioned matrices. The effectiveness of our proposed method and preconditioners are studied for solving some real-world examples represented by the heat conduction equation, the advection-diffusion equation, the wave equation and telegraph equations.

翻译：本文针对二维变系数进化型偏微分方程（PDEs），提出一种结合伯努利多项式与重心有理插值的数值方法：在时间方向上采用伯努利多项式，在空间方向上采用重心有理插值。我们证明了该数值格式的理论精度为$O\left((2\pi)^{-N}+h_x^{d_x-1}+h_y^{d_y-1}\right)$，其中$N$为时间基函数数目，$h_x$和$h_y$分别为$x$和$y$方向的网格尺度，且满足$0\leq d_x\leq \frac{b-a}{h_x}$, $0\leq d_y\leq\frac{d-c}{h_y}$。为高效求解离散化产生的相关线性系统，我们引入一类利用系数矩阵结构特性的维度扩展预处理器，并给出预处理矩阵特征值分布的理论分析。通过热传导方程、对流扩散方程、波动方程及电报方程等实际算例，验证了所提方法与预处理器的有效性。