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

中心格式可以作为通用有限差分方法，用于数值求解双曲守恒定律、哈密顿-雅可比方程和密切相关的对流-扩散方程。这样的方案与问题的特定特征结构无关，因此可以以一种直接的方式作为黑盒求解器来实现各种非线性方程，这些方程控制着大梯度现象的自发演化。一阶拉克斯-弗里德里希方案是这种中心方案的先驱。二阶Nessyahu-Tadmor格式提供了高分辨率，同时保留了riemann -free方法的简单性。在对流区，采用高阶分段多项式重构和高阶正交公式计算通量积分，提高了Nessyahu-Tadmor格式及其推广的分辨率。与此同时，当时间步长足够小时，这种交错中心方案家族会遭受过多的数值粘性。，由于存在（简并的）扩散项。最近，Kurganov和Tadmor引入了一个新的中心方案族，它保留了交错中心方案的简单性，但它们具有较小的数值粘性。特别地，这些方案承认一个简单的半离散公式。该项目旨在为守恒定律开发新的、最小耗散的全离散和半离散的中心方案。构建这些新方案背后的主要思想是使用更精确的局部传播速度信息，并根据其在不同大小的非对称黎曼风扇上的单元平均积分来实现（非光滑部分）近似解。双曲守恒定律、哈密顿-雅可比方程和对流-扩散方程具有重要的实际意义。它们控制着流体力学、气体动力学、磁流体动力学、天体物理学、地下水流动、气象学、半导体、反应流、油藏两相流、非牛顿流、锋面传播和其他几个领域中出现的各种物理现象。金融建模、交通流、微分对策、最优控制和图像增强是上述模型的最新应用。真正的多维高分辨率半离散中心方案为解决这些问题提供了一种相当简单和通用的方法。同时，中心方案的计算效率非常高。例如，最近的三维磁流体力学数值实验表明，与其他方法相比，使用中心方案可以更快地达到所需的分辨率约25倍。一般来说，当新的半离散中心方案用于解决实践中出现的复杂多维系统时，其优于其他迎风方法的优势被特别放大。所提出的方案还将应用于可压缩和不可压缩欧拉和纳维-斯托克斯方程、几何光学多相模型、多组分流和可压缩气泡模型、移动边界问题、非定常跨声速小扰动方程的冲击反射问题等重要问题。

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.