Volume 37, Issue 2
Effective Maximum Principles for Spectral Methods

Dong Li

Ann. Appl. Math., 37 (2021), pp. 131-290.

Published online: 2021-07

Export citation
  • 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.

  • AMS Subject Headings

35Q35

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

mpdongli@gmail.com (Dong Li)

  • BibTex
  • RIS
  • TXT
@Article{AAM-37-131, author = {Li , Dong}, title = {Effective Maximum Principles for Spectral Methods}, journal = {Annals of Applied Mathematics}, year = {2021}, volume = {37}, number = {2}, pages = {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.

}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2021-0003}, url = {http://global-sci.org/intro/article_detail/aam/19365.html} }
TY - JOUR T1 - Effective Maximum Principles for Spectral Methods AU - Li , Dong JO - Annals of Applied Mathematics VL - 2 SP - 131 EP - 290 PY - 2021 DA - 2021/07 SN - 37 DO - http://doi.org/10.4208/aam.OA-2021-0003 UR - https://global-sci.org/intro/article_detail/aam/19365.html KW - Spectral method, Allen-Cahn, maximum principle, Burgers, Navier-Stokes. AB -

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.

DongLi. (1970). Effective Maximum Principles for Spectral Methods. Annals of Applied Mathematics. 37 (2). 131-290. doi:10.4208/aam.OA-2021-0003
Copy to clipboard
The citation has been copied to your clipboard