单个守恒型方程熵耗散格式中熵耗散函数的构造

对于一维单个守恒律方程,文[8]设计了一种非线性守恒型差分格式.此格式为二阶Godunov型的,用的是分片线性重构(reconstruction),重构函数的斜率是根据熵耗散得到的.格式满足熵条件.与传统的守恒格式不同的是此格式在计算过程中不仅用到了数值解还用到了数值熵.在此格式中一个所谓的熵耗散函数起到了很重要的作用,它在每一个网格的计算中耗散熵,以保证格式满足熵条件.文[8]中设计的熵耗散函数比较复杂,并且不是很完善.故数值地分析了在格式的构造中为何应给熵以一定的耗散,及应耗散多少.并且给出了一个新的以数值解的二阶差分作为基本模块的熵耗散函数.最后给出了相应的数值算例.     

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.     

二维交错网格的GAUSS型格式

利用Gauss型求积公式在交错网格的情况下构造了一类不需解Riemann问题的求解二维双曲守恒律的二阶显式Gauss型差分格式,该格式在CFL条件限制下为MmB格式.并将格式推广到二维方程组,进行了数值试验.     

求解双曲守恒律方程的高分辨率熵相容格式

为提高熵相容格式的精度,利用限制器机制构造高分辨率格式,将构造的通量限制器插入熵相容格式,得到一类高分辨率熵相容格式.构造Euler方程高分辨率熵相容格式时,对熵相容格式中的几个参数做简单调整,提高了接触间断处的分辨率.将所得格式的数值结果与熵相容格式的数值结果比较表明,构造的高分辨率熵相容格式具有稳健和基本无振荡等特性.     

求解双曲型守恒律方程的一类自适应多分辨方法

针对双曲型守恒律方程问题,发展一种有效的自适应多分辨分析方法.通过对嵌套网格上的数值解构造离散多分辨分析,建立小波系数与多层嵌套网格点之间的对应关系.对于小波系数较大的网格点采用高精度WENO格式计算,其余区域则直接采用多项式插值.数值试验表明,该方法在保持原规则网格方法的精度和分辨率的同时,显著地减少计算的CPU时间.     

多尺度模拟中网格守恒重映算法

针对耦合微观分子动力学(MD)和宏观有限元方法(FE)的多尺度模拟,提出一类新的基于贡献单元法的网格守恒重映算法.由于物理量是由有限元节点以及相应区域的原子信息通过积分重构得到的,对结构和非结构网格都能适用.对于未知量定义在顶点的情形,引入辅助网格.数值例子验证了算法的准确性和有效性.     

一种基于HLLC算法的Mie-Grüneisen多介质混合模型

建立一种修正的HLLC （Harten-Lax-Van Leer-Contact）格式下稳定的Mie-Grüneisen多介质混合计算模型.Mie-Grüneisen混合模型中的通量包括守恒和非守恒两个部分，原始的HLLC格式对守恒部分适用，但是原始的HLLC格式直接用于非守恒部分，很难控制住数值振荡的产生.在原始格式中，间断面的移动速度为间断网格的左侧或右侧速度，修正后替换为网格中的平均速度，经过修正后，对HLLC格式重新进行推导，并随之扩展到二维问题.数值实验表明，利用修改后的HLLC格式，Mie-Grüneisen混合模型可以取得较好的稳定性和准确性.     

双曲守恒律的Taylor-Galerkin有限元方法

利用双曲守恒律的Hamilton-Jacobi方程形式,应用Taylor公式与Galerkin有限元给出了求解双曲守恒律的计算方法。采用TVD差分格式的构造思想,对数值通量作修正,在等距网格情形下有限元方法得到的计算格式满足TVD性质,并给出了数值例子。     

求解二维双曲型方程的一种基本守恒差分格式

构造了一种求解二维双曲型方程的基本守恒型差分格式,并证明了该格式的数值解是全变差有界的,在光滑区域具有二阶精度,按L¹范数及L^∞范数稳定,且其几乎处处有界收敛的极限解是微分方程的物理解。     

大长宽比扭曲网格上辅助网格差分方法

讨论抛物型方程的离散差分格式的构造,对九点差分格式进行了适用范围的讨论,并在此基础上提出辅助网格差分方法,用于处理因网格长宽比大且扭曲较大的网格引起的计算精度与计算效率降低的问题,该方法从守恒方程出发,将九点差分格式应用于按某种合适的方式进行重分之后的网格上,减少由于网格正则性差以及网格节点上的物理量采用周围网格量的加权平均等原因所引起的计算误差,得到一个新的但其解仍然逼近原来网格上的物理量的方程组.所构造的方法便于实施,且更适合于对实际物理模型的模拟,能比较好地适应流体大变形导致的网格扭曲,数值试验表明它有较好的数值精度和稳定性.     

Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method  and  for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.     

The discrete variational derivative method based on discrete differential forms

As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit this property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice. Lately, Furihata and Matsuo have developed the so-called “discrete variational derivative method” that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems. On the other hand, the theories of discrete differential forms have received much attention recently. These theories provide a discrete analogue of the vector calculus on general meshes. In this paper, we show that the discrete variational derivative method and the discrete differential forms by Bochev and Hyman can be combined. Applications to the Cahn–Hilliard equation and the Klein–Gordon equation on triangular meshes are provided as demonstrations. We also show that the schemes for these equations are H¹-stable under some assumptions. In particular, one for the nonlinear Klein–Gordon equation is obtained by combination of the energy conservation property and the discrete Poincaré inequality, which are the temporal and spacial structures that are preserved by the above methods.     

A high order Discontinuous Galerkin – Fourier incompressible 3D Navier–Stokes solver with rotating sliding meshes

We present the development of a sliding mesh capability for an unsteady high order (order ? 3) h/p Discontinuous Galerkin solver for the three-dimensional incompressible Navier–Stokes equations. A high order sliding mesh method is developed and implemented for flow simulation with relative rotational motion of an inner mesh with respect to an outer static mesh, through the use of curved boundary elements and mixed triangular–quadrilateral meshes.A second order stiffly stable method is used to discretise in time the Arbitrary Lagrangian–Eulerian form of the incompressible Navier–Stokes equations. Spatial discretisation is provided by the Symmetric Interior Penalty Galerkin formulation with modal basis functions in the x–y plane, allowing hanging nodes and sliding meshes without the requirement to use mortar type techniques. Spatial discretisation in the z-direction is provided by a purely spectral method that uses Fourier series and allows computation of spanwise periodic three-dimensional flows. The developed solver is shown to provide high order solutions, second order in time convergence rates and spectral convergence when solving the incompressible Navier–Stokes equations on meshes where fixed and rotating elements coexist.In addition, an exact implementation of the no-slip boundary condition is included for curved edges; circular arcs and NACA 4-digit airfoils, where analytic expressions for the geometry are used to compute the required metrics.The solver capabilities are tested for a number of two dimensional problems governed by the incompressible Navier–Stokes equations on static and rotating meshes: the Taylor vortex problem, a static and rotating symmetric NACA0015 airfoil and flows through three bladed cross-flow turbines. In addition, three dimensional flow solutions are demonstrated for a three bladed cross-flow turbine and a circular cylinder shadowed by a pitching NACA0012 airfoil.     

Locally implicit discontinuous Galerkin method for time domain electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes.     

A fast direct solver for elliptic problems on general meshes in 2D

We present a fast direct algorithm for solutions to linear systems arising from 2D elliptic equations. We follow the approach in Xia et al. (2009) on combining the multifrontal method with hierarchical matrices. We present a variant of that approach with additional hierarchical structure, extend it to quasi-uniform meshes, and detail an adaptive decomposition procedure for general meshes. Linear time complexity is shown for a quasi-regular grid and demonstrated via numerical results for the adaptive algorithm.     

基于近似Riemann解的有限体积ALE方法

研究二维平面坐标系和二维轴对称坐标系中四边形网格上可压缩流体力学的有限体积ALE(Arbitrary Lagrangian Eulerian)方法.数值方法采用节点中心有限体积法,数值通量采用适用于任意状态方程的HLLC(Harten-Lax-Van Leer-Collela)通量.空间二阶精度通过用WENO(weighted essentially non-oscillatory)方法对原始变量进行重构获得,时间离散采用两步显式Runge-Kutta格式.数值例子显示,方法具有良好的激波分辨能力和高精度的数值逼近能力.     

自适应间断有限元方法求解双曲守恒律方程

研究自适应Runge-Kutta间断Galerkin (RKDG)方法求解双曲守恒律方程组,并提出两种生成相容三角形网格的自适应算法.第一种算法适用于规则网格,实现简单、计算速度快.第二种算法基于非结构网格,设计一类基于间断界面的自适应网格加密策略,方法灵活高效.两种方法都具有令人满意的计算效果,而且降低了RKDG的计算量.     

Efficient spectral and pseudospectral algorithms for 3D simulations of whistler-mode waves in a plasma

Efficient spectral and pseudospectral algorithms for simulation of linear and nonlinear 3D whistler waves in a cold electron plasma are developed. These algorithms are applied to the simulation of whistler waves generated by loop antennas and spheromak-like stationary waves of considerable amplitude. The algorithms are linearly stable and show good stability properties for computations of nonlinear waves over tens of thousands of time steps. Additional speedups by factors of 10–20 (comparing single core CPU and one GPU) are achieved by using graphics processors (GPUs), which enable efficient numerical simulation of the wave propagation on relatively high resolution meshes (tens of millions nodes) in personal computing environment. Comparisons of the numerical results with analytical solutions and experiments show good agreement. The limitations of the codes and the performance of the GPU computing are discussed.     

Analysis of the monotonicity conditions in the mimetic finite difference method for elliptic problems

The maximum principle is one of the most important properties of solutions of partial differential equations. Its numerical analog, the discrete maximum principle (DMP), is one of the most difficult properties to achieve in numerical methods, especially when the computational mesh is distorted to adapt and conform to the physical domain or the problem coefficients are highly heterogeneous and anisotropic. Violation of the DMP may lead to numerical instabilities such as oscillations and to unphysical solutions such as heat flow from a cold material to a hot one. In this work, we investigate sufficient conditions to ensure the monotonicity of the mimetic finite difference (MFD) method on two- and three-dimensional meshes. These conditions result in a set of general inequalities for the elements of the mass matrix of every mesh element. Efficient solutions are devised for meshes consisting of simplexes, parallelograms and parallelepipeds, and orthogonal locally refined elements as those used in the AMR methodology. On simplicial meshes, it turns out that the MFD method coincides with the mixed-hybrid finite element methods based on the low-order Raviart–Thomas vector space. Thus, in this case we recover the well-established conventional angle conditions of such approximations. Instead, in the other cases a suitable design of the MFD method allows us to formulate a monotone discretization for which the existence of a DMP can be theoretically proved. Moreover, on meshes of parallelograms we establish a connection with a similar monotonicity condition proposed for the Multi-Point Flux Approximation (MPFA) methods. Numerical experiments confirm the effectiveness of the considered monotonicity conditions.