可压缩非牛顿流体力学方程组若干问题的研究

首先从可压缩非牛顿流体力学方程组研究的历史背景出发,以可压缩非牛顿流体力学方程组适定性研究为主线,通过介绍作者所在团队最近的相关工作,系统讲述了可压缩非牛顿流体力学方程组若干问题研究的新进展.     

二维可压缩流数值计算的严格误差估计

本文构造了计算二维可压缩流动的差分格式,严格估计了周期解问题的误差,并由此得到其收敛性.文中还严格估计了某一类初-边值问题的误差,并得到相应的收敛性.本文方法可推广应用于电磁流体力学方程组等.     

非结构网格上一种新型的高分辨率高精度无波动的有限元格式

在非结构网格上构造出无波动无自由参数耗散性有限元格式 ,即NND有限元格式 .通过若干个典型二维跨音速和超音速可压缩无粘定常流动的算例证明这确是一个高精度的 ,对激波具有高分辨率的无波动的新型有限元格式 .特别与网格自适应相结合 ,可得到十分满意的结果     

多项式基函数法

提出一种新型的数值计算方法--基函数法.此方法直接在非结构网格上离散微分算子,采用基函数展开逼近真实函数,构造出了导数的中心格式和迎风格式,取二阶多项式为基函数,并采用通量分裂法及中心格式和迎风格式相结合的技术以消除激波附近的非物理波动,构造出数值求解无粘可压缩流动二阶多项式的基函数格式,通过多个二维无粘超音速和跨音速可压缩流动典型算例的数值计算表明,该方法是一种高精度的、对激波具有高分辨率的无波动新型数值计算方法,与网格自适应技术相结合可得到十分满意的结果.     

拉格朗日乘子法的数学解释

本文简单回顾拉格朗日乘子法的产生的历史背景，并为其提供了一种新的数学解释.我们指出，拉格朗日乘子法将为求点到平面距离问题推广到了非线性规划求解问题.     

磁流体力学方程组的解——Dirac-Pauli表象的复变函数理论及其在流体力学中的应用(Ⅳ)

本文是文[1～3]的继续,在本文中(1) 我们将等熵可压缩无耗散的磁流体力学方程组化归为理想流体力学方程组的形式;应用文[3]的结果,我们可以得到磁流体力学推广的Chaplygin方程;从而,我们找到了关于这一类问题的通解.(2) 我们应用Dirac-Pauli表象的复变函数理论,将不可压缩磁流体力学的一般方程组化成关于流函数和"磁流函数"的两个非线性方程,并在有稳定磁场的条件下(即在运动粘性系数或粘流扩散系数等于磁扩散系数的条件下),求得了不可压缩磁流体力学方程组的精确稳定解.     

流体力学方程组的总熵增量小的守恒型差分格式

1.引言 近年来,国外许多学者对求解双曲守恒律组的高分辨率、高精度差分格式进行了深入的研究。例如MUSCL方法、TVD格式、PPM方法、各种限流的方法以及ENO格式等等。将这些方法应用于流体力学方程组,其数值实践的结果表明,在消除波后振荡、提高激波间断分辨率、提高计算精度等方面有明显的效果。在设计这些     

迎风紧致格式与驱动方腔流动问题的直接数值模拟

本文给出了一种求解不可压缩流动问题的高精度差分格式,即迎风紧致格式.出发方程采用二维非定常原始变量Naiver-Stokes方程组.在差分方程中,对流项采用三阶精度的迎风紧致差分,其余空间导数项采用四阶紧致差分.本文利用该差分格式在等距网格上数值模拟了驱动方腔流动中的分离涡运动.在257×257的细网格上,Re数最高计算到10000.Re≤5000时的计算结果与前人结果符合得很好.当Re≥7500时发现流动不存在定常层流解而为非定常周期性解,并首次给出了非定常解的结果。     

粘性流体力学的变分原理和广义变分原理

本文建立了不可压缩和可压缩粘性流体力学问题的变分原理,即最大功率消耗原理和它们的广义变分原理.     

功率型变分原理和功能型拟变分原理及其应用

自从钱伟长建立了功率型变分原理以来,功率型变分原理和功能型变分原理在理论方面和应用方面有什么区别和联系,成为学术界关注的课题.应用变积方法,根据Jourdain原理和d’Alembert原理,建立了不可压缩黏性流体力学的功率型变分原理和功能型拟变分原理,推导了不可压缩黏性流体力学的功率型变分原理的驻值条件和功能型拟变分原理的拟驻值条件.研究了不可压缩黏性流体力学的功率型变分原理在有限元素法中的应用.研究表明,功率型变分原理与Jourdain原理相吻合,功能型变分原理与d’Alembert原理相吻合.功率型变分原理直接在状态空间中研究问题,不仅在建立变分原理的过程中可以省略在时域空间中的一些变换,而且给动力学问题有限元素法的数值建模带来方便.     

A Survey of High Order Schemes for the Shallow Water Equations

In this paper, we survey our recent work on designing high order positivity-preserving well-balanced finite difference and finite volume WENO (weighted essentially non-oscillatory) schemes, and discontinuous Galerkin finite element schemes for solving the shallow water equations with a non-flat bottom topography. These schemes are genuinely high order accurate in smooth regions for general solutions, are essentially non-oscillatory for general solutions with discontinuities, and at the same time they preserve exactly the water at rest or the more general moving water steady state solutions. A simple positivity-preserving limiter, valid under suitable CFL condition, has been introduced in one dimension and reformulated to two dimensions with triangular meshes, and we prove that the resulting schemes guarantee the positivity of the water depth.     

High order positivity-preserving finite volume WENO schemes for a hierarchical size-structured population model

In this paper we develop high order positivity-preserving finite volume weighted essentially non-oscillatory (WENO) schemes for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. We carefully treat the technical complications in boundary conditions and global integration terms to ensure high order accuracy and the positivity-preserving property. Comparing with the previous high order difference WENO scheme for this model, the positivity-preserving finite volume WENO scheme has a comparable computational cost and accuracy, with the added advantages of being positivity-preserving and having L¹ stability. Numerical examples, including that of the evolution of the population of Gambusia affinis, are presented to illustrate the good performance of the scheme.     

High order positivity-preserving finite volume WENO schemes for a hierarchical size-structured population model

In this paper we develop high order positivity-preserving finite volume weighted essentially non-oscillatory (WENO) schemes for solving a hierarchical size-structured population model with nonlinear growth, mortality and reproduction rates. We carefully treat the technical complications in boundary conditions and global integration terms to ensure high order accuracy and the positivity-preserving property. Comparing with the previous high order difference WENO scheme for this model, the positivity-preserving finite volume WENO scheme has a comparable computational cost and accuracy, with the added advantages of being positivity-preserving and having $L^1$ stability. Numerical examples, including that of the evolution of the population of Gambusia affinis, are presented to illustrate the good performance of the scheme.     

A minimum entropy principle of high order schemes for gas dynamics equations

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy (Tadmor in Appl Numer Math 2:211–219, 1986). First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes (Khobalatte and Perthame in Math Comput 62:119–131, 1994) also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and Zhang et?al. (J Scientific Comput, in press), to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.     

Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

One‐dimensional models of gravity‐driven sedimentation of polydisperse suspensions with particles that belong to N size classes give rise to systems of N strongly coupled, nonlinear first‐order conservation laws for the local solids volume fractions. As the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic‐wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first‐order quasi‐linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so‐called Lagrangian‐remap [LR] schemes). This approach was advanced in (Bürger, Chalons, and Villada, SIAM J Sci Comput 35 (2013), B1341–B1368) for a multiclass Lighthill–Whitham‐Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian‐antidiffusive remap (L‐AR) schemes for the polydisperse sedimentation model is constructed. These L‐AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1109–1136, 2016     

A Vertex-Centered and Positivity-Preserving Finite Volume Scheme for Two-Dimensional Three-Temperature Radiation Diffusion Equations on General Polygonal Meshes

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multimaterial system and explain the exchange of energy among electrons, ions and photons. In this paper, we suggest a new positivity-preserving finite volume scheme for 2-D 3-T radiation diffusion equations on general polygonal meshes. The vertex unknowns are treated as primary ones for which the finite volume equations are constructed. The edge-midpoint and cell-centered unknowns are used as auxiliary ones and interpolated by the primary unknowns, which makes the final scheme a pure vertex-centered one. By comparison, most existing positivity-preserving finite volume schemes are cell-centered and based on the convex decomposition of the co-normal. Here, the co-normal decomposition is not convex in general, leading to a fixed stencil of the flux approximation and avoiding a certain search algorithm on complex grids. Moreover, the new scheme effectively alleviates the numerical heat-barrier issue suffered by most existing cell-centered or hybrid schemes in solving strongly nonlinear radiation diffusion equations. Numerical experiments demonstrate the second-order accuracy and the positivity of the solution on various distorted grids. For the problem without analytic solution, the contours of the numerical solutions obtained by our scheme on distorted meshes accord with those on smooth quadrilateral meshes.     

A virtual element method-based positivity-preserving conservative scheme for convection–diffusion problems on polygonal meshes

In this paper, we propose a positivity-preserving conservative scheme based on the virtual element method (VEM) to solve convection–diffusion problems on general meshes. As an extension of finite element methods to general polygonal elements, the VEM has many advantages such as substantial mathematical foundations, simplicity in implementation. However, it is neither positivity-preserving nor locally conservative. The purpose of this article is to develop a new scheme, which has the same accuracy as the VEM and preserves the positivity of the numerical solution and local conservation on primary grids. The first step is to calculate the cell-vertex values by the lowest-order VEM. Then, the nonlinear two-point flux approximations are utilized to obtain the nonnegativity of cell-centered values and the local conservation property. The new scheme inherits both advantages of the VEM and the nonlinear two-point flux approximations. Numerical results show that the new scheme can reach the optimal convergence order of the virtual element theory, that is, the second-order accuracy for the solution and the first-order accuracy for its gradient. Moreover, the obtained cell-centered values are nonnegative, which demonstrates the positivity-preserving property of our new scheme.     

A simple weighted essentially non-oscillatory limiter for the correction procedure via reconstruction (CPR) framework on unstructured meshes

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes on two-dimensional unstructured triangular meshes [39], to the correction procedure via reconstruction (CPR) framework for solving nonlinear hyperbolic conservation laws on two-dimensional unstructured triangular meshes with straight or curved edges. This is an extension of our earlier work [4] in which the WENO limiter was designed for the CPR framework on regular meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper uses information only from the target cell and its immediate neighbors. Hence, it is particularly simple to implement and will not harm the conservativeness and compactness of the CPR framework. Since the CPR framework with this WENO limiter does not in general satisfy the positivity preserving property, we also extend the positivity-preserving limiters [36], [33] to the CPR framework. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of this procedure.     

Power law Stokes equations with threshold slip boundary conditions: Numerical methods and implementation

For the power law Stokes equations driven by nonlinear slip boundary conditions of friction type, we propose three iterative schemes based on augmented Lagrangian approach and interior point method to solve the finite element approximation associated to the continuous problem. We formulate the variational problem which in this case is a variational inequality and construct the weak solution of the continuous problem. Next, we formulate two alternating direction methods based on augmented Lagrangian formalism in order to separate the velocity from the symmetric part the velocity gradient and tangential part of the velocity. Thirdly, we present some salient points of a path‐following variant of the interior point method associated to the finite element approximation of the problem. Some numerical experiments are performed to confirm the validity of the schemes and allow us to compare them.     

多边形网格上扩散方程新的单调格式

 本文在星形多边形网格上, 构造了扩散方程新的单调有限体积格式.该格式与现有的基于非线性两点流的单调格式的主要区别是, 在网格边的法向流离散模板中包含当前边上的点, 在推导离散法向流的表达式时采用了定义于当前边上的辅助未知量, 这样既可适应网格几何大变形, 同时又兼顾了当前网格边上物理量的变化. 在光滑解情形证明了离散法向流的相容性.对于具有强各向异性、非均匀张量扩散系数的扩散方程, 证明了新格式是单调的, 即格式可以保持解析解的正性. 数值结果表明在扭曲网格上, 所构造的格式是局部守恒和保正的, 对光滑解有高于一阶的精度, 并且, 针对非平衡辐射限流扩散问题, 数值结果验证了新格式在计算效率和守恒精度上优于九点格式.