非均质不可压缩纳维-斯托克斯方程有限差分法的数学分析

IF 2.1 2区 数学 Q1 MATHEMATICS, APPLIED
Kohei Soga
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引用次数: 0

摘要

本文对应用于有界域上不均匀(即密度和粘度非恒定)不可压缩纳维-斯托克斯系统的基本全离散有限差分法进行了数学分析。所提出的方法是经典方法,它由用于输运方程的 Lax-Friedrichs 显式方案和用于纳维-斯托克斯方程的 Ladyzhenskaya 隐式方案组成。在初始密度曲线严格远离零的条件下,该方案被证明在任意时间间隔内强收敛于一个弱解(直到子序列),这可以看作是系统弱解存在性的证明。这些结果包含一种新的奥宾-狮子-西蒙(Aubin-Lions-Simon)型紧凑性方法,该方法具有速度强规范与密度和速度乘积的弱规范之间的插值不等式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Mathematical analysis of a finite difference method for inhomogeneous incompressible Navier–Stokes equations

This paper provides mathematical analysis of an elementary fully discrete finite difference method applied to inhomogeneous (i.e., non-constant density and viscosity) incompressible Navier–Stokes system on a bounded domain. The proposed method is classical in the sense that it consists of a version of the Lax–Friedrichs explicit scheme for the transport equation and a version of Ladyzhenskaya’s implicit scheme for the Navier–Stokes equations. Under the condition that the initial density profile is strictly away from zero, the scheme is proven to be strongly convergent to a weak solution (up to a subsequence) within an arbitrary time interval, which can be seen as a proof of existence of a weak solution to the system. The results contain a new Aubin–Lions–Simon type compactness method with an interpolation inequality between strong norms of the velocity and a weak norm of the product of the density and velocity.

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来源期刊
Numerische Mathematik
Numerische Mathematik 数学-应用数学
CiteScore
4.10
自引率
4.80%
发文量
72
审稿时长
6-12 weeks
期刊介绍: Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers: 1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis) 2. Optimization and Control Theory 3. Mathematical Modeling 4. The mathematical aspects of Scientific Computing
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