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.

翻译：高阶非线性时间演化方程在科学与工程领域具有广泛的应用，例如在固体力学、材料科学和流体力学中。本文主要研究一种用于求解1至3阶时间相关微分方程的直接时间并行算法。不同于传统的时间步进方法，我们通过对时间离散化矩阵$B$进行对角化，直接从高阶演化方程求解“全一次性”系统。基于特征方程与切比雪夫多项式之间的联系，我们给出了$B$的特征向量矩阵$V$及其逆矩阵$V^{-1}$的显式公式。我们证明了$Cond_2\left( V \right) =\mathcal{O} \left( n^3 \right)$，其中$n$为时间步数。通过探索$B$的谱分解结构，设计了一种直接并行时间算法。数值实验展示了所提算法显著的计算加速效果。