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

中心格式可以作为数值求解双曲型守恒律方程、Hamilton-Jacobi方程以及与之密切相关的对流扩散方程的通用有限差分方法。这样的计划是不绑定到特定的本征结构的问题，因此可以在一个简单的方式实现作为黑盒求解器的各种非线性方程管理thesontaneous演变的大梯度现象。一阶Lax-Friedrichs格式是这类中心格式的先驱。二阶Nessyahu-Tadmor格式在保持Riemann无解方法简单性的同时提供了高分辨率。在对流区，通过高阶分段多项式重构和高阶求积公式计算通量积分，提高了Nessyahu-Tadmor格式及其推广的分辨率。同时，当施加足够小的时间步长时，这类交错中心格式遭受过度的数值粘性，例如，最近Kurganov和Tadmor提出了一类新的中心格式，它保持了交错中心格式的简单性，但数值粘性较小。特别是，theschemes承认一个简单的半离散配方。该项目的目的是开发新的，最小耗散的完全和半离散的中央计划的守恒律。这些新格式的主要思想是利用更精确的局部传播速度信息，并通过在不同大小的非对称Riemann扇上积分的单元平均来实现近似解（非光滑部分），双曲守恒律、Hamilton-Jacobi方程和对流扩散方程具有重要的实际意义。它们支配着出现在流体力学、气体动力学、磁流体动力学、天体物理学、地下水流、气象学、半导体、反应流、油藏中的两相流、非牛顿流、波前传播和其他几个领域中的各种物理现象。金融建模、交通流、微分对策、最优控制和图像增强等都是上述模型的最新应用，多维高分辨率半离散中心格式为解决这些问题提供了一种相当简单和通用的方法。同时，中心格式的计算效率也非常高。例如，最近在三维磁流体动力学的数值实验表明，使用中心的计划允许实现所需的分辨率约25倍的速度比其他方法。一般来说，新的半离散中心格式在求解实际中出现的复杂多维方程组时，其优越性尤其明显，它还可应用于可压缩和不可压缩的Euler和Navier-Stokes方程、几何光学多相模型、多组分流动和可压缩气泡模型、动边界问题、非定常跨音速小扰动方程的激波反射问题等。

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.