New High-Resolution Semi-Discrete Central Schemes: Derivation, Applications and Local Error Analysis

新的高分辨率半离散中心方案：推导、应用和局部误差分析

• 批准号: 0073631
• 负责人: Alexander Kurganov
• 金额: $ 6.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2001-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0073631&HistoricalAwards=false
• 关键词: New; Resolution; Semi; Discrete; Central

Central schemes may serve as universal finite-difference methodsfor numerically solving hyperbolic conservation laws, Hamilton-Jacobiequations and closely related convection-diffusion equations. Suchschemes are not tied to the specific eigen-structure of the problem,and hence can be implemented in a straightforward manner as black-boxsolvers for a wide variety of nonlinear equations governing thespontaneous evolution of large gradient phenomena. The first-order Lax-Friedrichs scheme is the forerunner for suchcentral schemes. The second-order Nessyahu-Tadmor scheme offers high resolution while retaining the simplicity of Riemann-solver-free approach. In the convective regime the improved resolution of the Nessyahu-Tadmor scheme and its generalizations is achieved by using high-order piecewise polynomial reconstructions and high-order quadrature formulas for computing the flux integrals. At the same time, this family of staggered central schemes suffers from excessivenumerical viscosity when a sufficiently small time step is enforced,e.g., due to the presence of (degenerate) diffusive term.Recently Kurganov and Tadmor introduced a new family of centralschemes, which retain the simplicity of staggered central schemes,yet they enjoy a smaller numerical viscosity. In particular, theseschemes admit a simple semi-discrete formulation. This project aims to develop new, minimally dissipative fully- and semi-discrete central schemes for conservation laws. The main ideas behind the construction of these new schemes is the use of more precise information of the localpropagation speed, and realizing the (non-smooth part of the) approximatesolution in terms of its cell averages integrated over the nonsymmetric Riemann fans of varying size.Hyperbolic conservation laws, Hamilton-Jacobi equations and convection-diffusion equations are of great practical importance. They govern avariety of physical phenomena that appear in fluid mechanics, gasdynamics, magnetohydrodynamics, astrophysics, groundwater flow,meteorology, semiconductors, reactive flows, two-phase flow in oilreservoirs, non-Newtonian flows, front propagation and several otherareas. Financial modeling, traffic flow, differential games, optimalcontrol and image enhancement are among the most recent applicationsof the above models.Genuinely multidimensional high-resolution semi-discrete central schemes provide a rather simple and universal method for solving these problems. At the same time, the computationalefficiency of central schemes is extremely high. For example, recent numerical experiments in three-dimensional magnetohydrodynamics demonstrate that using central schemes allows to achieve the desired resolution about 25 times faster in comparison with other methods. In general, the advantage of the new semi-discrete central schemes over alternative upwind methods is particularly amplified when they are used to solve complicated multidimensional systems arising in practice.The proposed schemes will be also applied to such important problemsas compressible and incompressible Euler and Navier-Stokes equations,multi-phase model of geometric optics, multicomponent flow andcompressible bubbles models, moving boundaries problems, shockreflection problem for the unsteady transonic small disturbanceequation and others.

中央方案可以用作通用有限差异方法，用于求解双曲线保护法，汉密尔顿 - 雅各布赛和密切相关的对流扩散方程。这些策略与问题的特定特征结构无关，因此可以直接地作为黑盒溶剂来实施，用于用于大量大梯度现象的各种非线性方程。一阶宽松液体方案是该中心方案的先驱。二阶Nessyahu-tadmor方案提供了高分辨率，同时保留了Riemann-Solver-Solver方法的简单性。在对流方案中，通过使用高阶分段多项式重建和高阶正交公式来计算磁通积分，可以改善Nessyahu-tadmor方案的分辨率及其概括。同时，当实施足够小的时间步长时，这种交错的中央计划的粘度过多，例如，由于存在（简并）差异术语，kurganov和tadmor介绍了一个新的中心策略家族，该家族退休了，这是一个新家族交错的中央计划的简单性，但它们的数值粘度较小。尤其是，这些策略允许一种简单的半污染公式。该项目旨在为保护法制定新的，最小化的全面耗散和半混凝土中央计划。这些新方案的构建背后的主要思想是使用局部传播速度的更精确的信息，并根据其细胞平均值（非对称的Riemann粉丝都具有不同大小的不同对称的Riemann粉丝，实现了（非平滑的部分）近似值。双曲线保护法，汉密尔顿 - 雅各比方程和对流扩散方程非常重要。它们控制着在流体力学，煤气动力学，磁动力学，天体物理学，地下水流量，气象学，半导体，反应性流动，反应性流动，非耐加工的两相流动，非牛顿顿流动，非牛顿流动，正面传播和几个其他因素中的宣誓体现象的贪婪。财务建模，交通流量，差异游戏，OptimalControl和图像增强是上述模型的最新应用之一。基础上多维的高分辨率半交流中央方案为解决这些问题提供了一种相当简单的通用方法。同时，中央方案的计算效率极高。例如，在三维磁性水力动力学中的最新数值实验表明，与其他方法相比，使用中央方案可以实现所需的分辨率约25倍。通常，在实践中出现的复杂的多维系统时，新的半分化中央方案比替代上风的方法的优势特别放大。所提出的方案也将应用于此类重要问题，可压缩和不可压缩的Euler和Navier -Stokes方程，几何光学的多相模型，多组分流量和可覆盖的气泡模型，移动边界问题，不稳定的跨性别小型干扰等方面的冲击反射问题等。