基于二阶格式的有限体积保正格式

从二阶线性格式出发，通过对法向通量进行重构，得到非线性两点通量，获得四面体网格上的单元中心型有限体积保正格式.该格式适用于求解间断和各向异性扩散系数问题.无需假设辅助未知量非负，避免了辅助未知量计算出负时"遇负置零"的人为处理方式；并且证明该格式在每个非线性Picard迭代步具有强保正性，即当源项和边界条件非负时，线性化格式的非平凡解是严格大于零的.数值算例验证该格式具有二阶收敛性且是保正的.     

球几何中子输运保正线性间断有限元格式

针对球几何中子输运方程线性间断有限元方法计算的负中子通量问题,构造了保正线性间断有限元格式,该格式保持中子角通量0阶矩和1阶矩。现有方法计算中子角通量非负时,采用传统的线性间断有限元方法,求解线性方程组;原方法计算出现负通量,则采用构造的保正格式,求解非线性方程组。编制了球几何中子输运问题保正格式程序模块,并集成到应用程序。数值算例表明构造的保正格式计算的中子通量非负,有效降低数值误差,提高数值计算的精度。     

非结构四边形网格下的一类保对称有限体元格式

针对定常扩散问题,在非结构四边形网格下,通过选取特殊的控制体和有限体元空间,建立两种保对称有限体元格式,在拟一致网格剖分下,当扩散系数光滑时,证明有限体元解函数在L²和H¹范数下均具有饱和误差阶.数值实验验证理论结果的正确性,同时表明新格式对扭曲大变形四边形网格、间断系数问题具有较强的适应性.在正交网格下,第二种格式对流(flux)函数在单元中心点的值还具有超逼近性.     

解三维扩散方程的积分内插法格式

研究三维扩散方程的数值模拟.在非正规六面体网格上,使用积分内插法建立扩散方程差分格式,涉及到27个相邻网格,适用于大变形网格上带间断系数的拟线性扩散方程的计算.叙述差分格式的建立,推导通量流和网格顶点温度的计算公式,给出了数值试验结果.     

基于特征理论的二维可压缩流动的二阶拉氏算法

提出一种求解二维拉氏可压缩流体力学方程的中心型二阶精度有限体积方法.利用特征理论构造网格节点处的局部近似演化算子,算子用来求解网格节点处的速度及压力,利用这些物理量更新节点位置及计算网格界面通量.通过结合一定的重构方案,该方法达到时、空二阶精度,并且形式简单、计算量小,适用于结构网格与非结构网格.典型数值实验表明,本文格式具有良好的收敛性、对称性及鲁棒性,且能自然地求解多物质流动问题.     

适应强扭曲网格的扩散方程时空二阶精度计算

以全局支撑算子方法为基础,通过引入面通量,构造了具有局部模板点的时空二阶精度格式。对于大变形扭曲网格,格式采用法向修正技术和合理的单元角体积计算方法,可以保持通量的精确性。算例表明该格式在非凸网格上能够精确获得线性解; 在非光滑网格上可以达到时空二阶精度; 能够较好地保持对称性; 并适合于三维非结构网格上的求解。     

基于节点重构的扩散方程有限体积格式

研究扭曲网格上扩散方程的九点格式.以经典的九点格式为基础,在通量连续条件下,构造出节点未知量一种新的计算方法,进而得到所希望的格式.分析及数值实验表明,在扭曲网格上,该格式对具有连续或间断扩散系数的问题能够保持较高的精度.     

含强间断导热系数的三维扩散方程数值计算

针对惯性约束聚变领域需要求解的含强间断导热系数的扩散方程,借鉴各向异性导热的思想,给出了一种导热系数定义在网格节点的支撑算子改进格式。改进后的方法保留了支撑算子格式的算子相容性优点。数值算例表明,新方法可以很好地处理含强间断和强非线性的热扩散问题,计算结果保持较高的精度,适合于辐射流体力学问题的模拟。     

基于非结构四边形网格的WENO格式

基于非结构四边形网格发展求解双曲守恒律的三阶加权基本无振荡（WENO）格式.针对任意非结构四边形网格选取重构模板，并给出基于线性多项式的三阶线性重构.但对于一般的非结构四边形网格，会出现非常大的线性权和负权，使得非线性重构的WENO格式对光滑问题也不稳定.本文给出一个处理非常大的线性权的优化重构方法，对优化后得到的负线性权采用分裂方法进行处理.对于非线性权，提出一种考虑局部网格和物理量间断的新光滑度量因子.采用优化重构方法和新的非线性权，当前的三阶WENO格式在质量很差的网格上也具有很好的稳定性.理论的三阶精度在数值精度测试算例中得到验证，同时一范数和无穷范数的误差绝对值不依赖于网格质量；具有强间断的数值结果证明了当前格式的有效性.     

求解多维欧拉方程的二阶非结构网格混合旋转Riemann求解器

将基于旋转近似Riemann求解器的二阶精度迎风型有限体积方法推广到非结构网格,采用基于网格中心的有限体积法,梯度的计算采用基于节点的方法引入更多的控制体模板,限制器的构造采用与非结构化网格相适应的形式.在求解Riemann问题时,沿具有一定物理意义的两个迎风方向,即控制体界面两侧速度差矢量方向及与之正交的方向.能够完全消除基于Riemann求解器的通量差分裂格式存在的激波不稳定或"红斑"现象.为减小计算量,采用HLL和Roe FDS混合旋转格式.     

各向异性扩散问题Kershaw格式的守恒保正修复算法

针对各向异性扩散方程Kershaw格式的数值解在正交网格及扭曲网格上计算出负的现象,给出一种守恒的保正修复算法（CENZ）,该算法对简单遇负置零（ENZ）方法进行改进,使修复后的数值解不仅具有非负性,而且保持法向通量的局部守恒性.数值算例表明,该方法不受计算网格类型和扩散系数各向异性比的限制,可用于对任意违背单调性（或保正性）的有限体积格式数值解的修复.     

Linearity preserving nine-point schemes for diffusion equation on distorted quadrilateral meshes

In this paper, we employ the so-called linearity preserving method, which requires that a difference scheme should be exact on linear solutions, to derive a nine-point difference scheme for the numerical solution of diffusion equation on the structured quadrilateral meshes. This scheme uses firstly both cell-centered unknowns and vertex unknowns, and then the vertex unknowns are treated as a linear combination of the surrounding cell-centered unknowns, which reduces the scheme to a cell-centered one. The weights in the linear combination are derived through the linearity preserving approach and can be obtained by solving a local linear system whose solvability is rigorously discussed. Moreover, the relations between our linearity preserving scheme and some existing schemes are also discussed, by which a generalized multipoint flux approximation scheme based on the linearity preserving criterion is suggested. Numerical experiments show that the linearity preserving schemes in this paper have nearly second order accuracy on many highly skewed and highly distorted structured quadrilateral meshes.     

The finite volume scheme preserving extremum principle for diffusion equations on polygonal meshes

We construct a new nonlinear finite volume scheme for diffusion equation on polygonal meshes and prove that the scheme satisfies the discrete extremum principle. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results are presented to show how our scheme works for preserving discrete extremum principle and positivity on various distorted meshes.     

An improved monotone finite volume scheme for diffusion equation on polygonal meshes

We construct a new nonlinear monotone finite volume scheme for diffusion equation on polygonal meshes. The new scheme uses the cell-edge unknowns instead of cell-vertex unknowns as the auxiliary unknowns in order to improve the accuracy of monotone scheme. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes. Specially, numerical results show that the new scheme is robust, and more accurate than the existing monotone scheme on some kinds of meshes.     

Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes

We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients.     

Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations

One of the main challenges in computational simulations of gas detonation propagation is that negative density or negative pressure may emerge during the time evolution, which will cause blow-ups. Therefore, schemes with provable positivity-preserving of density and pressure are desired. First order and second order positivity-preserving schemes were well studied, e.g., [6], [10]. For high order discontinuous Galerkin (DG) method, even though the characteristicwise TVB limiter in [1], [2] can kill oscillations, it is not sufficient to maintain the positivity. A simple solution for arbitrarily high order positivity-preserving schemes solving Euler equations was proposed recently in [22]. In this paper, we first discuss an extension of the technique in [22], [23], [24] to design arbitrarily high order positivity-preserving DG schemes for reactive Euler equations. We then present a simpler and more robust implementation of the positivity-preserving limiter than the one in [22]. Numerical tests, including very demanding examples in gaseous detonations, indicate that the third order DG scheme with the new positivity-preserving limiter produces satisfying results even without the TVB limiter.     

On maximum-principle-satisfying high order schemes for scalar conservation laws

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization solving one-dimensional scalar conservation laws. Strong stability preserving (SSP) high order time discretizations will keep the maximum principle. It is straightforward to extend the method to two and higher dimensions on rectangular meshes. We also show that the same limiter can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. Numerical tests for both the WENO finite volume scheme and the DG method are reported.     

Positivity-preserving high order finite difference WENO schemes for compressible Euler equations

In Zhang and Shu (2010) [20], Zhang and Shu (2011) [21] and Zhang et al. (in press) [23], we constructed uniformly high order accurate discontinuous Galerkin (DG) and finite volume schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) finite difference schemes for compressible Euler equations. General equations of state and source terms are also discussed. Numerical tests of the fifth order finite difference WENO scheme are reported to demonstrate the good behavior of such schemes.     

多流管网格上九点格式的节点插值方法

多流管方法是二维多介质辐射流体力学数值模拟中一类常用的求解方法,它采用Lagrange-Euler混合型四边形网格,称为多流管网格。通常其网格品质高于一般的四边形网格。在这类网格上,可以利用网格特性对九点扩散格式中的节点插值方法进行改进。本文利用调和平均点和梯度离散构造的方法提出几种节点插值方法。并给出数值实验,说明现有应用程序中的节点插值方法损失精度,而新的节点插值方法能够使得九点格式在多流管网格上具有二阶精度。