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

TL;DR

This paper introduces a direct parallel-in-time algorithm for higher-order nonlinear time-evolution equations, leveraging spectral decomposition and Chebyshev polynomials to achieve significant computational speedups.

Contribution

It develops a novel direct time-parallel method for orders 1 to 3 evolution equations using spectral diagonalization, differing from traditional time-stepping approaches.

Findings

The algorithm achieves substantial computational speedup.

Explicit formulas for eigenvectors are derived.

The condition number of the eigenvector matrix grows as O(n^3).

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.