谱方法的有效最大值原理
Effective maximum principles for spectral methods
摘要：
许多物理问题如Allen-Cahn流都有自然的最大值原理，它可以根据边界数据、初始条件和算子系数对物理解进行强逐点控制。在连续问题中具有根本重要性的尖锐/严格最大值原理在数值离散化下往往不成立。以往的研究主要集中在设计优良的数值格式，特别是对非线性问题，保持尖锐的最大值原理。然而，这些尖锐的原则，不仅有时会引入不必要的严格条件的数值方案，而且完全离开了许多强大的基于频率的方法无人看管，很少直接在尖锐的最大范数拓扑分析。一个突出的例子是加权残差方法家族中的谱方法。在这项工作中，我们介绍和发展一个新的框架几乎尖锐的最大值原理，允许数值解偏离尖锐的界限由一个可控的离散化误差：我们称之为有效的最大值原理。我们展示了经典的傅里叶谱方法的分析，包括傅里叶伽辽金和傅里叶配置在空间与前向欧拉在时间或二阶Strang分裂。模型方程包括双势Allen-Cahn方程、Burgers方程和Navier-斯托克斯方程。我们在非常一般的参数条件下给出了有效的最大值原理的一个全面的证明。
英文摘要：
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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