Effective maximum principles for spectral methods

TL;DR

This paper develops a new framework called effective maximum principles for spectral methods, allowing controlled deviations from sharp maximum bounds in numerical solutions of PDEs like Allen-Cahn, Burgers, and Navier-Stokes equations.

Contribution

It introduces the concept of effective maximum principles for spectral methods, providing a rigorous analysis that accommodates discretization errors while preserving maximum norm bounds.

Findings

Effective maximum principles are established for Fourier spectral methods.

The framework applies to various PDEs including Allen-Cahn, Burgers, and Navier-Stokes.

The analysis covers different time discretizations like forward Euler and Strang splitting.

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.