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Effective Maximum Principles for Spectral Methods

Effective Maximum Principles for Spectral Methods
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Year:    2021

Author:    Dong Li

Annals of Applied Mathematics, Vol. 37 (2021), Iss. 2 : pp. 131–290

Abstract

Many physical problems such as Allen-Cahn flows have natural maximum principles which yield strong point-wise control of the physical solutions in terms of the boundary data, the initial conditions and the operator coefficients. Sharp/strict maximum principles insomuch of fundamental importance for the continuous problem often do not persist under numerical  discretization. A lot of past research concentrates on designing fine numerical schemes which preserves the sharp maximum principles especially for nonlinear problems. However, these sharp principles  not only sometimes introduce unwanted stringent conditions on the numerical schemes but also completely leaves  many powerful frequency-based methods unattended and rarely analyzed directly in the sharp maximum norm topology. A prominent example is the spectral methods in the family of weighted residual methods.

In this work we introduce and develop a new framework of almost sharp maximum principles which allow the numerical solutions to deviate from the sharp bound by a controllable discretization error: we call them effective maximum principles. We showcase the analysis for the classical Fourier spectral methods including Fourier Galerkin and Fourier collocation in space with forward Euler in time or second order Strang splitting.  The model equations include the Allen-Cahn equations with double well potential, the Burgers equation and the Navier-Stokes equations. We give a comprehensive proof of the effective maximum principles under very general parametric conditions.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aam.OA-2021-0003

Annals of Applied Mathematics, Vol. 37 (2021), Iss. 2 : pp. 131–290

Published online:    2021-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    160

Keywords:    Spectral method Allen-Cahn maximum principle Burgers Navier-Stokes.

Author Details

Dong Li

  1. On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation

    Zhang, Hong | Zhang, Gengen | Liu, Ziyuan | Qian, Xu | Song, Songhe

    Advances in Computational Mathematics, Vol. 50 (2024), Iss. 3

    https://doi.org/10.1007/s10444-024-10143-6 [Citations: 0]

  2. A family of structure-preserving exponential time differencing Runge–Kutta schemes for the viscous Cahn–Hilliard equation

    Sun, Jingwei | Zhang, Hong | Qian, Xu | Song, Songhe

    Journal of Computational Physics, Vol. 492 (2023), Iss. P.112414

    https://doi.org/10.1016/j.jcp.2023.112414 [Citations: 5]

  3. Why large time-stepping methods for the Cahn-Hilliard equation is stable

    Li, Dong

    Mathematics of Computation, Vol. (2022), Iss.

    https://doi.org/10.1090/mcom/3768 [Citations: 2]

  4. Stability and convergence of Strang splitting. Part II: Tensorial Allen-Cahn equations

    Li, Dong | Quan, Chaoyu | Xu, Jiao

    Journal of Computational Physics, Vol. 454 (2022), Iss. P.110985

    https://doi.org/10.1016/j.jcp.2022.110985 [Citations: 8]

  5. Remarks on the Bernstein inequality for higher order operators and related results

    Li, Dong | Sire, Yannick

    Transactions of the American Mathematical Society, Vol. (2022), Iss.

    https://doi.org/10.1090/tran/8708 [Citations: 0]

  6. A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials

    Li, Dong

    Discrete & Continuous Dynamical Systems, Vol. 42 (2022), Iss. 5 P.2453

    https://doi.org/10.3934/dcds.2021198 [Citations: 4]

  7. A maximum principle of the Fourier spectral method for diffusion equations

    Kim, Junseok | Kwak, Soobin | Lee, Hyun Geun | Hwang, Youngjin | Ham, Seokjun

    Electronic Research Archive, Vol. 31 (2023), Iss. 9 P.5396

    https://doi.org/10.3934/era.2023273 [Citations: 1]

  8. A maximum bound principle preserving iteration technique for a class of semilinear parabolic equations

    Gong, Yuezheng | Ji, Bingquan | Liao, Hong-lin

    Applied Numerical Mathematics, Vol. 184 (2023), Iss. P.482

    https://doi.org/10.1016/j.apnum.2022.11.002 [Citations: 4]

  9. Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation

    Li, Dong | Quan, Chaoyu | Tang, Tao

    Mathematics of Computation, Vol. 91 (2021), Iss. 334 P.785

    https://doi.org/10.1090/mcom/3704 [Citations: 11]

  10. Stability and convergence of Strang splitting. Part I: Scalar Allen-Cahn equation

    Li, Dong | Quan, Chaoyu | Xu, Jiao

    Journal of Computational Physics, Vol. 458 (2022), Iss. P.111087

    https://doi.org/10.1016/j.jcp.2022.111087 [Citations: 17]

  11. Temporal high-order, unconditionally maximum-principle-preserving integrating factor multi-step methods for Allen-Cahn-type parabolic equations

    Zhang, Hong | Yan, Jingye | Qian, Xu | Song, Songhe

    Applied Numerical Mathematics, Vol. 186 (2023), Iss. P.19

    https://doi.org/10.1016/j.apnum.2022.12.020 [Citations: 8]

  12. Large time-stepping, delay-free, and invariant-set-preserving integrators for the viscous Cahn–Hilliard–Oono equation

    Zhang, Hong | Liu, Lele | Qian, Xu | Song, Songhe

    Journal of Computational Physics, Vol. 499 (2024), Iss. P.112708

    https://doi.org/10.1016/j.jcp.2023.112708 [Citations: 1]