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.     

基于紧致差分格式的高效时域有限差分算法

探讨一种基于紧致差分格式的高效时域有限差分算法(high-order compact-FDTD),该方法不仅提高计算精度,而且网格结点少、内存使用率和CPU时间大为降低.利用紧致格式FDTD方法实现无耗波导系统及光子晶体光纤中电磁波传播的数值模拟.通过计算实例验证算法的高效性.     

poisson方程的预示校正差分格式

本文提出求解二维、三维Poisson方程的Dirichlet和Neumann两类边值问题的预示校正差分格式。它完整地包括求解区域的内结点格式、边界面结点格式、边界线结点格式和边界角结点格式。这种新的差分格式达到四阶精度,并可通过对选择因子的优越使计算误差达到最小。     

一维非线性对流占优扩散方程的变网格特征差分方法

针对一维非线性对流占优扩散方程,提出了一类变网格特征差分格式,该格式能够根据解的梯度变化及时对计算网格进行调整.与均匀网格格式相比,给出的变网格特征差分格式对于对流占优扩散问题有着更好的计算效果.     

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。     

基于中心差分的对流扩散方程四阶紧凑格式

在经典中心差分格式的基础上,提出对流扩散方程的四阶紧凑差分格式。具体方法是,先就一维情形,将中心差分格式改造为不受网格Reynolds数限制的恒稳二阶格式,再在不增加相关网格点的前提下,通过格式中对流系数和源项的摄动处理,使稳格式的精度提高至四阶。本文并作一、二、三维流动模型方程及高Rayleigh数自然对流传热问题的数值求解,例示本文格式的优良性态。     

四阶紧致差分格式对静力模式中垂直网格计算频散性的影响

为了说明四阶紧致差分格式在大气和海洋数值模式中的潜在价值,提出一种通用方法,推导静力线性斜压适应方程组在微分和差分情况下的频散关系,水平尺度分100 km,10 km和1 km三种情况,从频率、水平群速和垂直群速方面,对采用二阶中央差和四阶紧致差分格式情况下,非跳点网格(N网格)、Lorenz网格(L网格)、Charney-Phillips网格(CP网格)、Lorenz时间跳点网格(LTS网格)和Charney-Phillips时间跳点网格(CPTS网格)的计算特性进行比较,发现采用高精度的四阶紧致差分格式总体上可以明显减少上述三种水平尺度波动在N网格、CP网格、L网格和CPTS网格上的频率、水平群速和垂直群速误差,但对LTS网格,采用四阶紧致差分格式,会使得计算水平群速和垂直群速误差变大.     

三维不可压N-S方程的多重网格求解

应用全近似存储(Full Approximation Storage,FAS)多重网格法和人工压缩性方法求解了三维不可压Navi-er-Stokes方程.在解粗网格差分方程时,对Neumann边界条件采用增量形式进行更新,离散方程用对角化形式的近似隐式因子分解格式求解,其中空间无粘项分别用MUSCL格式和对称TVD格式进行离散.对90°弯曲的方截面管道流动和4:1椭球体层流绕流的数值模拟表明,多重网格的计算时间比单重网格节省一半以上,且无限制函数的MUSCL格式比TVD格式对流动结构有更好的分辨能力.     

磁流体方腔槽道流整体线性稳定性数值方法研究

将Chebyshev谱配置法和基于非均匀网格的高阶FD-q差分格式运用于磁流体方腔槽道流整体线性稳定性研究,比较两类数值方法的优缺点.Chebyshev谱配置法收敛快且精度高,但需要构造非常庞大的满矩阵,存储量和计算开销巨大;高阶FD-q差分格式采用了基于Kosloff-Tal-Ezer变换的Chebyshev谱配置点作为离散网格,在保持较高网格收敛精度的同时,差分格式可以采用稀疏矩阵进行存储,显著降低了存储量和计算开销.相比传统的谱配置法,基于非均匀网格的高阶FD-q差分格式计算效率得到显著的提升,将高阶FD-q差分格式运用于非正则模线性最优瞬态增长的计算,计算效果良好.     

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.     

Robust,multidimensional mesh-motion based on Monge–Kantorovich equidistribution

Mesh-motion (r-refinement) grid adaptivity schemes are attractive due to their potential to minimize the numerical error for a prescribed number of degrees of freedom. However, a key roadblock to a widespread deployment of this class of techniques has been the formulation of robust, reliable mesh-motion governing principles, which (1) guarantee a solution in multiple dimensions (2D and 3D), (2) avoid grid tangling (or folding of the mesh, whereby edges of a grid cell cross somewhere in the domain), and (3) can be solved effectively and efficiently. In this study, we formulate such a mesh-motion governing principle, based on volume equidistribution via Monge–Kantorovich optimization (MK). In earlier publications [1], [2], the advantages of this approach with regard to these points have been demonstrated for the time-independent case. In this study, we demonstrate that Monge–Kantorovich equidistribution can in fact be used effectively in a time-stepping context, and delivers an elegant solution to the otherwise pervasive problem of grid tangling in mesh-motion approaches, without resorting to ad hoc time-dependent terms (as in moving-mesh PDEs, or MMPDEs [3], [4]). We explore two distinct r-refinement implementations of MK: the direct method, where the current mesh relates to an initial, unchanging mesh, and the sequential method, where the current mesh is related to the previous one in time. We demonstrate that the direct approach is superior with regard to mesh distortion and robustness. The properties of the approach are illustrated with a hyperbolic PDE, the advection of a passive scalar, in 2D and 3D. Velocity flow fields with and without flow shear are considered. Three-dimensional grid, time-step, and nonlinear tolerance convergence studies are presented which demonstrate the optimality of the approach.     

重分网格技巧初探

本文讨论按等距原则重新划分拉氏网格的技巧,应用重分阈值η和四步重分方法。新、旧网格之间,按体积份额守恒地分配有关网格量,数值试验表明,为了提高重分效果,不仅需要调整重分区域内部网格点的位置,而且还要适当调整边界网格点的位置。     

加权ENO格式的构造及数值模拟

将加权ENO格式推广到非结构三角形网格上,构造了一类加权ENO有限体积格式,提出的插值多项式的构造方式,可以减少计算时间.对于出现的病态方程组,给出了解决方法.此外还给出了插值点的选取方式及加权因子的构造方法.结合三阶TVD Runge Kutta时间离散,对二维欧拉方程组进行了数值试验.     

基于移动网格的熵稳定格式

提出一种基于移动网格的熵稳定格式求解双曲型守恒律方程.该方法利用等分布原理得到新的网格分布,基于守恒型插值公式计算新的网格上的物理量,使用熵稳定数值通量和三阶强稳定Runge-Kutta时间推进方法得到下一时刻的数值解.数值算例表明该格式不仅能有效提高解在间断处的分辨率,而且能消除可能产生的伪振荡.     

Double-families of quasi-positive Darcy-flux approximations with highly anisotropic tensors on structured and unstructured grids

This paper focuses on flux-continuous pressure equation approximation for strongly anisotropic media. Previous work on families of flux-continuous schemes for solving the general geometry–permeability tensor pressure equation has focused on single-parameter families. These schemes have been shown to remove the O(1) errors introduced by standard two-point flux reservoir simulation schemes when applied to full-tensor flow approximation. Improved convergence of the schemes has also been established for specific quadrature points. However these schemes have conditional M-matrices depending on the strength of the off-diagonal tensor coefficients. When applied to cases involving full-tensors arising from strongly anisotropic media, the point-wise continuous schemes can fail to satisfy the maximum principle and induce severe spurious oscillations in the numerical pressure solution.New double-family flux-continuous locally conservative schemes are presented for the general geometry–permeability tensor pressure equation. The new double-family formulation is shown to expand on the current single-parameter range of existing conditional M-matrix schemes. The conditional M-matrix bounds on a double-family formulation are identified for both quadrilateral and triangle cell grids. A quasi-positive QM-matrix analysis is presented that classifies the behaviour of the new schemes with respect to double-family quadrature in regions beyond the M-matrix bounds. The extension to double-family quadrature is shown to be beneficial, resulting in novel optimal anisotropic quadrature schemes. The new methods are applied to strongly anisotropic full-tensor field problems and yield results with sharp resolution, with only minor or practically zero spurious oscillations.     

Coupled-channel R-matrix method on a Lagrange mesh

The coupled-channel R-matrix method on a Lagrange mesh is a very simple approximation of the R-matrix method with a basis. The mesh points are zeros of shifted Legendre polynomials. Bound-state energies and scattering matrices are easily calculated with small numbers of potential values at mesh points. A test with an exactly solvable two-channel potential provides an excellent accuracy over a broad energy range with only 30 mesh points. The efficiency of the method is illustrated for a single channel on α + α scattering and for two channels on the deuteron ground-state energy and on nucleon-nucleon scattering.     

Modeling merging and breakup in the moving mesh interface tracking method for multiphase flow simulations

The three-dimensional, moving mesh interface tracking (MMIT) method coupled with local mesh adaptations by Quan and Schmidt [S.P. Quan, D.P. Schmidt, A moving mesh interface tracking method for 3D incompressible two-phase flows, J. Comput. Phys. 221 (2007) 761–780] demonstrated the capability to accurately simulate multiphase flows, to handle large deformation, and also to perform interface pinch-off for some specific cases. However, another challenge, i.e. how to handle interface merging (such as droplet coalescence) has not been addressed. In this paper, we present a mesh combination scheme for interface connection and a more general mesh separation algorithm for interface breakup. These two schemes are based on the conversion of liquid cells in one phase to another fluid by changing the fluid properties of the cells in the combination or separation region. After the conversion, the newly created interface is usually ragged, and a local projection method is employed to smooth the interface. Extra mesh adaptation criteria are introduced to handle colliding interfaces with almost zero curvatures as the distance between the interfaces diminishes. Simulations of droplet pair collisions including both head-on and off-center coalescences show that the mesh adaptations are capable of resolving very small length scales, and the mesh combination and mesh separation schemes can handle the topological transitions in multiphase flows. The potential of our method to perform detailed investigations of droplet coalescence and breakup is also displayed.     

Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number

We propose a theoretical framework to clearly explain the inaccuracy of Godunov type schemes applied to the compressible Euler system at low Mach number on a Cartesian mesh. In particular, we clearly explain why this inaccuracy problem concerns the 2D or 3D geometry and does not concern the 1D geometry. The theoretical arguments are based on the Hodge decomposition, on the fact that an appropriate well-prepared subspace is invariant for the linear wave equation and on the notion of first-order modified equation. This theoretical approach allows to propose a simple modification that can be applied to any colocated scheme of Godunov type or not in order to define a large class of colocated schemes accurate at low Mach number on any mesh. It also allows to justify colocated schemes that are accurate at low Mach number as, for example, the Roe–Turkel and the AUSM⁺-up schemes, and to find a link with a colocated incompressible scheme stabilized with a Brezzi–Pitkäranta type stabilization. Numerical results justify the theoretical arguments proposed in this paper.