粘性机制下一类熵相容格式的对比研究

研究了一种人工和物理耗散机制下的离散熵相容格式,探讨数值粘性和物理粘性的大小以及它们所起的作用.所得结论是:在激波捕捉的过程中,粘性系数越大,则无需加入人工粘性项;粘性系数较小时,除了物理粘性项,还需要加入人工粘性项来得到熵相容格式.首先研究了一维粘性Burgers方程离散熵相容格式,再将其推广至Navier-Stokes方程.数值算例采用空间半离散格式,并结合显式三步三阶Runge-Kutta(RK3)方法进行时间推进.这两类方程的数值结果表明,最终选取的熵相容格式能够准确地捕捉到激波.     

气体动力学燃烧模型的广义Riemann问题

考察了在(x,t)平面上原点(t＞0)的邻域内气体动力学燃烧模型的广义Riemann问题．在改进的熵条件下构造了此问题的唯一解．它们是自相似ZND燃烧模型的极限．发现对某些情形,广义Riemann问题的解与相应的Riemann问题的解有本质的不同．特别地,扰动会使得相应Riemann问题的强爆轰波转化为由预压激波点燃的弱爆燃波．在一些情形,尽管相应的Riemann解中不含燃烧波,扰动后燃烧波会出现．这反映了未燃气体的不稳定性．     

平面平行管道中的MHD方程组的边界层

该文研究平面平行管道中不可压缩MHD方程组的边界层问题.利用多尺度分析和精细的能量方法,证明了当粘性系数与磁耗散系数趋近于0时,粘性与磁耗散MHD方程组的解收敛到理想MHD方程组的解.     

相变演化Suliciu模型中波的相互作用

描述相变演化的Suliciu模型，其基本波可由行波分析得到．对于任何给定分两段常值的初始状态，相应的Riemann解是某些基本波的组合．对分三段常值的初始状态，解由上述Piemann解构成，其中相邻两状态间以基本波连接．当基本波发生碰撞时，新的Riemann问题形成．通过研究这些Riemann。问题，可以在适当的参数空间中对基本波之间复杂的相互作用加以分类．     

Eikonal型方程粘性解的表达式

对于圆锥型和棱锥型Hamiltonian的Eikonal型方程,本文给出了一种几何方法,得出其初值问题解的表达式并且说明由此式给出的解为原初值问题的粘性解．首先用一个凸函数序列逼近Eikonal型方程中的Hamiltonian,再由Hopf-Lax公式给出方程序列的粘性解,最后证明了该粘性解序列会收敛到Eikonal方程的粘性解．     

一种守恒型间断跟踪法在一维单守恒律方程非凸流上的实现

本文的第二作者在近几年发展了一种守恒型的间断跟踪法，该跟踪法是以解的守恒性质作为跟踪的机制，而不是象传统的跟踪法利用Rankine-Hugoniot条件来进行跟踪．本文中主要研究将该算法推广至单个守恒律非凸流的情况．利用精确求解Riemann问题，我们很好地处理了非凸流Riemann解的激波和稀疏波的并存结构，既实现了对激波的跟踪，又成功地分辨出稀疏波．第四节的数值例子。显示了满意的数值结果．     

关于实轴上多解析函数Riemann边值问题的可解性

本文研究了实轴上具有不同因子的多解析函数的Riemann边值问题的可解性．利用所谓的转化法．建立了Riemann问题的可解性与其相联问题的解之间的关系。该结果推广了解析函数的相应理论。     

带非局部耗散项的守恒律方程大扰动解的整体存在性

本文考虑带非局部耗散项的单个守恒律方程大扰动解的整体存在性.首先,针对方程次临界和临界两种不同的情形,利用Green函数方法和环形分解的技巧,构造开放式高频估计方法,得到了一个新的解的正则性准则.然后,利用极大值原理得到方程解的极大模的有界性,验证了次临界情形下解满足相应的正则性准则.对于临界这一更困难的情形,本文应用非线性极大值原理方法得到了更好一点的有界性估计,验证了临界情形下解也满足相应的正则性准则,从而得到了Cauchy问题大扰动经典解的整体存在性.     

具有障碍的二阶完全非线性椭圆型弱藕合方程组的粘性解

本文考虑具有障碍的二阶完全非线性椭圆型弱藕合方程组的粘性解．在自然结构条件之下,利用罚技巧,增生算子方法和摄动理论,证明了粘性解的存在性,与Person方法相比,本文的方法的优点在于不依赖于上下解的存在性。最后,给出了粘性解唯一的充分条件．     

带有摩擦项的广义Chaplygin气体非对称Keyfitz-Kranzer方程组的Riemann问题

研究了带有摩擦项的广义Chaplygin气体非对称Keyfitz-Kranzer方程组的Riemann问题,并得到其Riemann解的整体结构.Riemann解中包含激波,稀疏波,接触间断和δ-激波.与齐次非对称Keyfitz-Kranzer方程组不同的是非齐次非对称Keyfitz-Kranzer方程组的Riemann解是非自相似的.     

基于子网格边界近似Riemann解的人为粘性方法

在交错网格型Lagrange(拉格朗日)流体力学算法中,通常采用人为粘性捕捉激波,人为粘性的好坏对于计算结果至关重要.研究了一种基于子网格边界处近似Riemann解的新型人为粘性.新人为粘性能够满足动量守恒和熵不等式.利用子网格边界速度差中引入的限制器,新人为粘性能够区分激波和等熵压缩,并能满足球对称问题中的波面不变性.新人为粘性在典型模型数值模拟及惯性约束聚变黑腔整体数值模拟中取得了较好的结果.     

A New Version of Iterative Method for Solving Riemann Problem

A new version of iterative method for solving Riemann problem of gas dynamics is presented. In practice the new procedure exhibited a good convergence in cases where Riemann solution involves a strong rarefaction wave or two rarefaction waves. In the other cases the new version is identical with Godunov procedure.     

Nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws

We consider the Riemann problem for a system of two decoupled, nonstrictly hyperbolic, Burgers-like conservation equations with added artificial viscosity. We analytically establish two different vanishing viscosity limits for the solution of this system, which correspond to the two cases where one of the viscosities vanishes much faster than the other. This is done without altering the initial condition as is necessary with travelling wave methods. Numerical evidence is then provided to show that when the two viscosities vanish at the same rate, the solution converges to a limit that lies strictly between the two previously established limits. Finally, we use control theory to explain the mechanism behind this nonuniqueness behavior, which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur.     

MUSTA: A multi-stage numerical flux

Numerical methods for conservation laws constructed in the framework of finite volume and discontinuous Galerkin finite elements require, as the building block, a monotone numerical flux. In this paper we present some preliminary results on the MUSTA approach [E.F. Toro, Multi-stage predictor–corrector fluxes for hyperbolic equations, Technical Report NI03037-NPA, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, 17th June, 2003] for constructing upwind numerical fluxes. The scheme may be interpreted as an un-conventional approximate Riemann solver that has simplicity and generality as its main features. When used in its first-order mode we observe that the scheme achieves the accuracy of the Godunov method used in conjunction with the exact Riemann solver, which is the reference first-order method for hyperbolic systems. At least for the scalar model hyperbolic equation, the Godunov scheme is the best of all first-order monote schemes, it has the smallest truncation error. Extensions of the scheme of this paper are realized in the framework of existing approaches. Here we present a second-order TVD (TVD for the scalar case) extension and show numerical results for the two-dimensional Euler equations on non-Cartesian geometries. The schemes find their best justification when solving very complex systems for which the solution of the Riemann problem, in the classical sense, is too complex, too costly or is simply unavailable.     

Computation of discontinuous solutions of fluid dynamics equations with entropy nondecrease guarantee

A new formulation of the Godunov scheme with linear Riemann problems is proposed that guarantees a nondecrease in entropy. The new version of the method is described for the simplest example of one-dimensional gas dynamics in Lagrangian coordinates.     

二维双曲守恒律的大时间步长Godunov方法(英文)

考虑了关于二维守恒律的大时间步长Godunov方法.该方法是关于一维问题的自然推广.证明了文中给出的数值流函数下,该方法是守恒的.进一步还给出了近似Riemann解应满足的条件,并且证明了利用满足这些条件的近似Riemann解的大时间步长Godunov方法守恒.最后,补充证明了满足这些条件的近似Riemann解是满足熵条件的.     

Numerical studies to propose a ghost particle removed SPH (GR-SPH) method

This paper aims to propose a new modified SPH method with novel treatments at the boundaries. Although SPH methods decrease contradictions due to grid distortions compared to traditional mesh-based methods, the penetration of particles through boundaries and the consistency problem make the simulation of problems with definite boundaries a concern. The use of ghost boundary particles and the insertion of artificial forces at the boundaries are the most popular boundary consistency treatments proposed thus far. The use of artificial forces causes the mixing of molecular and finite theories, which can violate the conservation of momentum. This paper shows how the use of ghost boundary particles can violate the continuity equation in problems with non-zero velocity divergence. This study proposes a novel ghost particle removed SPH (GR-SPH) method that discards all ghost particles and artificial forces at the boundaries. Liner layers and liner particles have been defined inside the domain instead of ghost boundary particles in such a way that the so-called violations can partially be remedied. Based on the continuity equation and kernel function unity specification, a novel truncation correction factor has been defined for density renormalization to override the consistency problem at the boundaries. In addition, a new method is proposed to detect the particles near complex wall boundaries and evaluate the normal distance from boundaries. Finally, some benchmark problems have been solved to show the capabilities of the new modified SPH method for the prediction of both particle location and pressure distribution with acceptable accuracy. The GR-SPH method facilitates programming, with fewer particles contributing to the computations. Comparison of its outcomes with published results shows that the new treatments executed at the boundaries are effective.     

A MUSCL smoothed particles hydrodynamics for compressible multi‐material flows

By incorporating the Monotone Upwind Scheme of Conservation Law (MUSCL) scheme into the smoothed particles hydrodynamics (SPH) method and making use of an interparticle contact algorithm, we present a MUSCL–SPH scheme of second order for multifluid computations, which extends the Riemann‐solved‐based SPH method. The numerical tests demonstrate high accuracy and resolution of the scheme for both shocks, contact discontinuities, and rarefaction waves in the one‐dimensional shock tube problem. For the two‐dimensional cylindrical Noh and shock‐bubble interaction problems, the MUSCL–SPH scheme can resolve shocks well. Copyright © 2015 John Wiley & Sons, Ltd.     

三维PTT黏弹性液滴撞击固壁面问题的改进SPH模拟

基于光滑粒子动力学(smoothed particle hydrodynamics, SPH)方法,对三维Phan-Thien Tanner(PTT)黏弹性液滴撞击固壁面问题进行了数值模拟.为了有效地防止粒子穿透固壁,且缩减三维数值模拟所消耗的计算时间,提出了一种适合三维数值模拟的改进固壁边界处理方法.为了消除张力不稳定性问题,采用一种简化的人工应力技术.应用改进SPH方法对三维PTT黏弹性液滴撞击固壁面问题进行了数值模拟,精细地捕捉了液滴在不同时刻的自由面,讨论了PTT黏弹性液滴不同于Newton(牛顿)液滴的流动特征,分析了PTT拉伸参数对液滴宽度、高度和弹性收缩比等的影响.模拟结果表明,改进SPH方法能够有效而准确地描述三维PTT黏弹性液滴撞击固壁面问题的复杂流变特性和自由面变化特征.     

The relaxation schemes for systems of conservation laws in arbitrary space dimensions

We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solvers temporally. Numerical results for 1-D and 2-D problems are presented. The second-order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. ©1995 John Wiley & Sons, Inc.