非正交网格上的九点格式在热传导问题中的改进

对文献[1]在二维Lagrange流体力学网格上构造的扩散方程九点差分格式做了进一步的讨论和改进,给出了一般形式边界条件的计算格式。数值试验的结果表明,这些改进提高了原格式数值结果的精度。     

基于Voronoi网格的扩散方程差分格式

在Voronoi网格上利用一种基于回路积分法的有限体积法构造扩散方程的的差分格式.在这种特殊的网格上离散扩散方程比通常在四边形网格上离散的格式要简单,不会引进角点未知量,提高了对网格边上的流的离散精度,及差分格式整体精度.这种Voronoi网格上的扩散计算也可以与单元中心流体力学计算耦合.数值算例表明这种格式比四边形网格上的格式精度高,且能更好的应对网格扭曲情形.     

九点格式中网格边上切向流的计算——节点自适应加权格式

针对网格扭曲的不同情形,直接考虑网格边上切向流的离散.基于扩散方程法向流连续的条件,给出离散法向流的构造,导出扭曲网格上九点计算格式中网格边上离散切向流的表达式,从而推导出加权系数的计算公式,适应于各种扭曲的网格.数值结果表明,与九点格式中节点量简单加权的方法相比,基于网格边离散切向流的节点自适应加权九点格式的精度有明显改进,迭代求解次数减少,计算效率明显提高.     

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.     

三维对流扩散方程非等距网格上的四阶紧致格式及其多重网格方法

提出了数值求解三维变系数对流扩散方程非等距网格上的四阶精度19点紧致差分格式,为了提高求解效率,采用多重网格方法求解高精度格式所形成的大型代数方程组。数值实验结果表明本文方法对于不同的网格雷诺数问题,在精确性、稳定性和减少计算工作量方面均明显优于7点中心差分格式。     

二维半导体问题在复合网格上的有限差分格式

半导体器件的瞬时状态由3个方程组成的非线性偏微分方程组的初边值问题决定.依据实际数值模拟的需要,提出了一类二维半导体问题在时空局部加密复合网格上的有限差分形式,电场位势方程、电子和空穴浓度方程分别用五点差分格式和修正迎风格式近似,且在交界面上采用线性插值,并给出了电子和空穴浓度的最大模误差估计,最后给出了数值算例.     

振动NACA0012翼型的移动网格数值模拟

利用流体力学方程的积分形式给出非结构移动网格上离散格式,利用自适应移动网格方法移动网格,进而得到网格速度.对振动Naca0012翼型问题,分三种类型确定网格速度,再结合Riemann问题的解法器构造数值通量,得到移动网格单元上新的物理量.数值实验表明这种格式同时具有高效、高分辨率的特点.     

基于变分原理的二维热传导方程差分格式

研究二维热传导方程的差分数值模拟.用变分原理在不规则结构网格上建立热流通量形式的差分格式.将热流通量作为未知函数求泛函极值,并与温度函数联立求解.克服通常九点格式用插值方法计算网格边界上的热传导系数和网格结点上的温度所引入的误差.     

热传导方程的一类无网格方法

构造求解热传导方程的一类无网格方法,只要选择好每个节点的适当的邻点集合,便可利用节点信息顺利进行计算.作为特殊情形,也可在各种结构或非结构网格上进行计算.在矩形或均匀平行四边形网格上进行计算时具有二阶精度,当在任意的不规则四边形或三角形网格上计算时仍然是守恒的和相容的,且至少具有一阶精度.作为数值试验,将该方法用于在不规则四边形网格上及四边形与三角形混合网格上求解二维非线性抛物型方程,并在不规则四边形网格上求解二维三温辐射热传导方程,均获得了较为理想的数值结果.     

基于WENO重构的高阶移动网格动理学格式

提出一种基于WENO重构的高阶(至少三阶)移动网格动理学格式.利用流体力学方程的积分形式得到移动网格上离散格式,再利用自适应移动网格方法移动网格,进而得到网格速度,利用WENO重构得到高阶插值多项式,最后使用时间方向上精确的动理学数值方法构造数值通量,得到移动网格单元上新的物理量.数值实验表明这种格式同时具有高精度、高分辨率的特点.     

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.     

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。     

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.     

Fully HOC Scheme for Mixed Convection Flow in a Lid-Driven Cavity Filled with a Nanofluid

A fully higher-order compact (HOC) finite difference scheme on the 9-point two-dimensional (2D) stencil is formulated for solving the steady-state laminar mixed convection flow in a lid-driven inclined square enclosure filled with water-$Al_2O_3$ nanofluid. Two cases are considered depending on the direction of temperature gradient imposed (Case I, top and bottom; Case II, left and right). The developed equations are given in terms of the stream function-vorticity formulation and are non-dimensionalized and then solved numerically by a fourth-order accurate compact finite difference method. Unlike other compact solution procedure in literature for this physical configuration, the present method is fully compact and fully higher-order accurate. The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and averaged Nusselt number. Comparisons with previously published work are performed and found to be in excellent agreement. A parametric study is conducted and a set of graphical results is presented and discussed to elucidate that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by inclination of the enclosure at moderate and large Richardson numbers.     

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.     

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.     

Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady Navier–Stokes equations

The fluid dynamic equations are discretized by a high-order spectral volume (SV) method on unstructured tetrahedral grids. We solve the steady state equations by advancing in time using a backward Euler (BE) scheme. To avoid the inversion of a large matrix we approximate BE by an implicit lower–upper symmetric Gauss–Seidel (LU-SGS) algorithm. The implicit method addresses the stiffness in the discrete Navier–Stokes equations associated with stretched meshes. The LU-SGS algorithm is then used as a smoother for a p-multigrid approach. A Von Neumann stability analysis is applied to the two-dimensional linear advection equation to determine its damping properties. The implicit LU-SGS scheme is used to solve the two-dimensional (2D) compressible laminar Navier–Stokes equations. We compute the solution of a laminar external flow over a cylinder and around an airfoil at low Mach number. We compare the convergence rates with explicit Runge–Kutta (E-RK) schemes employed as a smoother. The effects of the cell aspect ratio and the low Mach number on the convergence are investigated. With the p-multigrid method and the implicit smoother the computational time can be reduced by a factor of up to 5–10 compared with a well tuned E-RK scheme.