A Bernoulli-barycentric rational matrix collocation method with preconditioning for a class of evolutionary PDEs
<|reference_start|>A Bernoulli-barycentric rational matrix collocation method with preconditioning for a class of evolutionary PDEs: 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.<|reference_end|>
@article{luo2024a,
    title={A Bernoulli-barycentric rational matrix collocation method with
  preconditioning for a class of evolutionary PDEs},
    author={Wei-Hua Luo, Xian-Ming Gu, Bruno Carpentieri, Jun Guo},
    journal={arXiv preprint arXiv:2402.03861},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.03861},
    primaryClass={math.NA cs.NA}
}