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：针对二维变系数演化型偏微分方程，本文提出一种结合时间域Bernoulli多项式与空间域重心有理插值的Bernoulli-重心有理矩阵配置法。证明了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$N$为时间基函数数量，$h_x$与$h_y$分别为$x$与$y$方向网格尺寸，且$0\leq d_x\leq \frac{b-a}{h_x},~0\leq d_y\leq\frac{d-c}{h_y}$。为高效求解离散化产生的线性系统，引入一类利用系数矩阵结构特性的维度扩展型预处理器，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。