A Bernoulli-barycentric rational matrix collocation method with preconditioning for a class of evolutionary PDEs

TL;DR

The paper develops a Bernoulli-barycentric rational matrix-collocation approach to solving two-dimensional evolutionary PDEs with variable coefficients, achieving high-order temporal and spatial accuracy. By combining Bernoulli polynomials in time with barycentric rational interpolants in space, it yields a structured linear system with a clear error bound $O\left((2\pi)^{-N}+h_x^{d_x-1}+h_y^{d_y-1}\right)$. To address computational cost, it introduces dimension-expanded preconditioners $P_DE1, P_DE2, P_DE3$ that exploit the coefficient matrix structure and improve Krylov subspace convergence, with proven eigenvalue distribution properties. Numerical experiments on heat conduction, advection-diffusion, wave, and telegraph equations demonstrate superior accuracy and efficiency compared with conventional methods, validating the approach for practical PDE problems and motivating extensions to nonlinear cases.

Abstract

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 $O\left((2π)^{-N}+h_x^{d_x-1}+h_y^{d_y-1}\right)$ 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.

Figures (6)

• Figure 1: Sparsity pattern plots of the coefficient matrix $\tilde{H}$ ($\beta_1=0,~\beta_2=1$) in (\ref{['10yue30_1']}) (left), the augmented matrix $\mathbb{H}$ (middle) and the preconditioner $P_{\rm DE1}$ (right).
• Figure 2: Eigenvalue distributions of the preconditioned matrix $P_{\rm DE1}^{-1}\mathbb{H}$ and original matrix $\mathbb{H}$ in Example 3 under $8\times 8$ and $12\times 12$ uniform grids in space, respectively.
• Figure 3: Sparsity pattern plots of the coefficient matrix $H$ ($\beta_1=1,~\beta_2=0$) in (\ref{['1yue19_2']}) (left), the augmented matrix $\hat{\mathbb{H}}$ (middle) and the preconditioner $P_{\rm DE2}$ (right).
• Figure 4: Eigenvalue distributions of the preconditioned matrix $P_{\rm DE2}^{-1}\hat{\mathbb{H}}$ and original matrix $\hat{\mathbb{H}}$ in Example 4 under $8\times 8$ and $12\times 12$ uniform grids in space, respectively.
• Figure 5: Sparsity pattern plots of the coefficient matrix $H$ ($\beta_1\neq0,~\beta_2\neq0$) in (\ref{['1yue19_3']}) (left), the augmented matrix $\tilde{\mathbb{H}}$ (middle) and the preconditioner $P_{\rm DE3}$ (right).

Theorems & Definitions (25)

• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Theorem 3.1
• Proof 1
• Remark 3.2
• Theorem 3.3
• Theorem 3.4