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

TL;DR

This work addresses the challenge of time-parallel solution for higher-order nonlinear time-evolution equations by a direct PinT approach that diagonalizes the time-discretization matrix $B$, yielding an all-at-once system $M=(B\otimes I_x+I_t\otimes A)$. The authors derive explicit spectral data for $B$, proving $B=VDV^{-1}$ with eigenvalues $\lambda_j=i x_j$ where $x_j$ are roots of a Chebyshev-based polynomial, and show $Cond_2\left( V \right)=O\left(n^3\right)$. They then develop a fast $O(n^2)$ algorithm for computing $V^{-1}$ and a fast $O(n)$ method for the spectral decomposition of $B$ via Newton iterations on a Chebyshev-related root equation. Numerical experiments demonstrate substantial speedups over traditional eigendecomposition, validating the practicality of the method for nonlinear first- to third-order problems and highlighting its potential impact on large-scale wave, fluid, and PDE simulations. Together, these results offer a scalable, direct time-parallel framework for high-order nonlinear evolution equations with concrete algorithms for diagonalization and inversion.

Abstract

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.

Figures (4)

• Figure 1: Comparison of the CPU time with different number of cores of Example 1, where $m=512^{2}$ and $N_t = 2^3, \cdots, 2^9$.
• Figure 2: Comparison of the CPU time with different number of cores of Example 2, where $m=512^{2}$ and $N_t = 2^3, \cdots, 2^9$.
• Figure 3: Comparison of the CPU time with different number of cores of Example 3, where $m=512^{2}$ and $N_t = 2^3, \cdots, 2^9$.
• Figure 4: Comparison of the CPU time with different number of cores of Example 4, where $m=512^{2}$ and $N_t = 2^3, \cdots, 2^9$.

Theorems & Definitions (12)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof