arXiv subjects

Qian-Yu Shu

Publications and source records attributed to Qian-Yu Shu.

3 recordsLinked to original sources

arXiv2025

A second-order generalized BDF method for the two-dimensional (modified) Fisher-Kolmogorov-Petrovsky-Piskunov equation

The Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) equation is a classical reaction-diffusion equation with broad applications such as biology, chemistry and physics. In this paper, an alternative second-order scheme is proposed by employing a shifted BDF2 method to approximate the two-dimensional (modified) Fisher-KPP equation. We both consider an uniform and a nonuniform time steps of such the scheme. The stability of the uniform discretization scheme is proved. Numerical experiments demonstrate that our uniform and non-uniform schemes are robust and accurate.

math.NA↗

arXiv2025

A direct PinT algorithm for higher-order nonlinear time-evolution equations

Higher-order nonlinear time-evolution equations have widespread applications in science and engineering, such as in solid mechanics, materials science, and fluid mechanics. This paper mainly studies a direct time-parallel algorithm for solving time-dependent differential equations of orders 1 to 3. Different from the traditional time-stepping approach, we directly solve the all-at-once system from higher-order evolution equations by diagonalization the time discretization matrix $B$. Based on the connection between the characteristic equation and Chebyshev polynomials, we give explicit formulas for the eigenvector matrix $V$ of $B$ and its inverse $V^{-1}$. We prove that $Cond_2\left( V \right) =\mathcal{O} \left( n^3 \right)$, where $n$ is the number of time steps. A direct parallel-in-time algorithm is designed by exploring the structure of the spectral decomposition of $B$. Numerical experiments are provided to show the significant computational speedup of the proposed algorithm.

math.NA↗

arXiv2023

Minimax programming problems subject to addition-Łukasiewicz fuzzy relational inequalities and their optimal solutions

This article focuses on minimax programming problems subject to addition-Łukasiewicz fuzzy relational inequalities. We first establish two necessary and sufficient conditions that a solution of the fuzzy relational inequalities is a minimal one and explore the existence condition of the unique minimal solution. We also supply an algorithm to search for minimal solutions of the fuzzy relational inequalities starting from a given solution. We then apply minimal solutions of the fuzzy relational inequalities to the minimax programming problems for searching optimal solutions. We provide two algorithms to solve a kind of single variable optimization problems, and obtain the greatest optimal solution. The algorithm for finding minimal solutions of a given solution are also used for searching minimal optimal solutions.

math.GM↗