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| Effective Maximum Principles for Spectral Methods |
| Dong Li |
| (Department of Mathematics and SUSTech International Center for
Mathematics, Southern University of Science and Technology,
Shenzhen 518055, Guangdong, China) |
| DOI： |
| Abstract: |
| Many physical problems such as Allen-Cahn flows have natural max-
imum 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 max-
imum 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 meth-
ods 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 e ective
maximum principles under very general parametric conditions. |
| Key words: Spectral method, Allen-Cahn, maximum principle, Burgers, Navier-Stokes. |