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.
翻译:Allen-Cahn流动等许多物理问题都有自然的最高原则,这些自然最高原则在边界数据、初始条件和操作系数方面对实际解决办法产生强有力的点控制。对于持续问题具有根本重要性的尖锐/严格最大原则往往不会在数字分解的情况下持续存在。过去的许多研究都集中于设计细微数字计划,保留最严格的最高原则,特别是针对非线性问题。然而,这些尖锐原则不仅有时对数字方案提出不必要的严格条件,而且完全使许多强大的基于频率的方法得不到注意,而且很少直接在最高标准表层中分析。一个突出的例子就是加权残余方法组群中的光谱法。在这个工作中,我们引入和开发了一个几乎最敏锐的最高原则的新框架,使数字解决办法偏离了可控离散错误的尖锐界限:我们称之为有效的最高原则。我们展示了典型的四面光谱方法的分析,包括Fourier Galerkin和Fourier的频率共位同地在时间或第二顺序上与Euler的距离。模型方程式包括Allen-Cahn-Cahrang Pal 等式模型方程式,在普遍正方程式中提供了有效的双方程式。