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

2025/07/08 by Zhong, Shun-Zhi, Zhao, Yong-Liang, Shu, Qian-Yu
#FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2507.05743

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 Cond₂( V ) =O ( n³ ), 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.

Citations

• Time parallelization for hyperbolic and parabolic problems
• An optimal diagonalization-based preconditioner for parabolic optimal control problems
• Diagonalization-Based Parallel-in-Time Preconditioners for Instationary Fluid Flow Control Problems
• A preconditioned MINRES method for optimal control of wave equations and its asymptotic spectral distribution theory
• Diagonalization-based preconditioners and generalized convergence bounds for ParaOpt
• On generalized preconditioners for time-parallel parabolic optimal control
• A parallel-in-time preconditioner for Crank-Nicolson discretization of a parabolic optimal control problem
• A well-conditioned direct PinT algorithm for first-and second-order evolutionary equations
• A parallel-in-time collocation method using diagonalization: theory and implementation for linear problems
• Space-time block preconditioning for incompressible flow
• Matrix equation techniques for certain evolutionary partial differential equations
• A Note on Parallel Preconditioning for All-At-Once Evolutionary PDEs
• Stable parareal in time method for first and second order hyperbolic system

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