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.
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