arXiv subjects

Xiangyi Peng

Publications and source records attributed to Xiangyi Peng.

4 recordsLinked to original sources

arXiv2026

Temporal Two-Grid Compact Difference Scheme for Benjamin-Bona-Mahony-Burgers Equation

This paper proposes a temporal two-grid compact difference (TTCD) scheme for solving the Benjamin-Bona-Mahony-Burgers (BBMB) equation with initial and periodic boundary conditions. The method consists of three main steps: first, solving a nonlinear system on a coarse time grid of size $\tau_c$; then obtaining a coarse approximation on the fine time grid of size $\tau_f$ via linear Lagrange interpolation; and finally solving a linearized scheme on the fine grid to obtain the corrected solution. The TTCD scheme reduces computational cost without sacrificing accuracy. Moreover, using the energy method, we rigorously prove the conservation property, unique solvability, convergence, and stability of the proposed scheme. It is shown that the method achieves convergence of order $\mathcal{O}(\tau_c^2 + \tau_f^2 + h^4)$ in the maximum norm, where $h$ is space step size. Finally, some numerical experiments are provided to demonstrate the effectiveness and feasibility of the proposed strategy.

math.NA↗

arXiv2025

An Efficient Space-Time Two-Grid Compact Difference Scheme for the Two-Dimensional Viscous Burgers' Equation

This work proposes an efficient space-time two-grid compact difference (ST-TGCD) scheme for solving the two-dimensional (2D) viscous Burgers' equation subject to initial and periodic boundary conditions. The proposed approach combines a compact finite difference discretization with a two-grid strategy to achieve high computational efficiency without sacrificing accuracy. In the coarse-grid stage, a fixed-point iteration is employed to handle the nonlinear system, while in the fine-grid stage, linear temporal and cubic spatial Lagrange interpolations are used to construct initial approximations. The final fine-grid solution is refined through a carefully designed linearized correction scheme. Rigorous analysis establishes unconditional convergence of the method, demonstrating second-order accuracy in time and fourth-order accuracy in space. Numerical experiments verify the theoretical results and show that the ST-TGCD scheme reduces CPU time by more than 70\% compared with the traditional nonlinear compact difference (NCD) method, while maintaining comparable accuracy. These findings confirm the proposed scheme as a highly efficient alternative to conventional nonlinear approaches.

math.NA↗

arXiv2025

High-order discretization errors for the Caputo derivative in H\"{o}lder spaces

Building upon the recent work of Teso and Plociniczak (2025) regarding L1 discretization errors for the Caputo derivative in H\"{o}lder spaces, this study extends the analysis to higher-order discretization errors within the same functional framework. We first investigate truncation errors for the L2 and L1-2 methods, which approximate the Caputo derivative via piecewise quadratic interpolation. Then we generalize the results to arbitrary high-order discretization. Theoretical analyses reveal a unified error structure across all schemes: the convergence order equals the difference between the smoothness degree of the function space and the fractional derivative order, i.e., order of error = degree of smoothness - order of the derivative. Numerical experiments validate these theoretical findings.

math.NA↗

arXiv2022

Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers' equation

In this paper, based on the developed nonlinear fourth-order operator and method of order reduction, a novel fourth-order compact difference scheme is constructed for the mixed-type time-fractional Burgers' equation, from which $L_1$-discretization formula is employed to deal with the terms of fractional derivative, and the nonlinear convection term is discretized by nonlinear compact difference operator. Then a fully discrete compact difference scheme can be established by approximating spatial second-order derivative with classic compact difference formula. The convergence and stability are rigorously proved in the $L^{\infty}$-norm by the energy argument and mathematical induction. Finally, several numerical experiments are provided to verify the theoretical analysis.

math.NA↗