A New Class of High-Order Energy Stable Flux Reconstruction Schemes for Triangular Elements

The flux reconstruction (FR) approach allows various well-known high-order schemes, such as collocation based nodal discontinuous Galerkin (DG) methods and spectral difference (SD) methods, to be cast within a single unifying framework. Recently, the authors identified a new class of FR schemes for 1D conservation laws, which are simple to implement, efficient and guaranteed to be linearly stable for all orders of accuracy. The new schemes can easily be extended to quadrilateral elements via the construction of tensor product bases. However, for triangular elements, such a construction is not possible. Since numerical simulations over complicated geometries often require the computational domain to be tessellated with simplex elements, the development of stable FR schemes on simplex elements is highly desirable. In this article, a new class of energy stable FR schemes for triangular elements is developed. The schemes are parameterized by a single scalar quantity, which can be adjusted to provide an infinite range of linearly stable high-order methods on triangular elements. Von Neumann stability analysis is conducted on the new class of schemes, which allows identification of schemes with increased explicit time-step limits compared to the collocation based nodal DG method. Numerical experiments are performed to confirm that the new schemes yield the optimal order of accuracy for linear advection on triangular grids.     

On the Non-linear Stability of Flux Reconstruction Schemes

The flux reconstruction (FR) approach unifies various high-order schemes, including collocation based nodal discontinuous Galerkin (DG) methods, and all spectral difference methods (at least for a linear flux function), within a single framework. Recently a new range of linearly stable FR schemes have been identified, henceforth referred to as Vincent-Castonguay-Jameson-Huynh (VCJH) schemes. In this short note non-linear stability properties of FR schemes are elucidated via analysis of linearly stable VCJH schemes (so as to focus attention solely on issues of non-linear stability). It is shown that linearly stable VCJH schemes (at least in their standard form) may be unstable if the flux function is non-linear. This instability is due to aliasing errors, which manifest since FR schemes (in their standard form) utilize a collocation projection at the solution points to construct a polynomial approximation of the flux. Strategies for minimizing such aliasing driven instabilities are discussed within the context of the FR approach. In particular, it is shown that the location of the solution points will have a significant effect on non-linear stability. This result is important, since linear analysis of FR schemes implies stability is independent of solution point location. Finally, it is shown that if an exact L2 projection is employed to construct an approximation of the flux (as opposed to a collocation projection), then aliasing errors and hence aliasing driven instabilities will be eliminated. However, performing such a projection exactly, or at least very accurately, would be more costly than performing a collocation projection, and would certainly impact the inherent efficiency and simplicity of the FR approach. It can be noted that in all above regards, non-linear stability properties of FR schemes are similar to those of nodal DG schemes. The findings should motivate further research into the non-linear performance of FR schemes, which have hitherto been developed and analyzed solely in the context of a linear flux function.     

A New Class Of Adams-Bashforth Schemes For Odes

We consider a new class of Adams-Bashforth schemes for solving first order differential equations. The method of collocation was used. In the process of the derivation, we observed that it is possible to obtain the discrete scheme from the continuous scheme by collocating at a given point. Also, it is noticed that the discrete solutions and continuous solution are the same at the grid points.     

High-Order Well-Balanced Finite Difference WENO Schemes for a Class of Hyperbolic Systems with Source Terms

In this paper, we generalize the high order well-balanced finite difference weighted essentially non-oscillatory (WENO) scheme, designed earlier by us in Xing and Shu (2005, J. Comput. phys. 208, 206–227) for the shallow water equations, to solve a wider class of hyperbolic systems with separable source terms including the elastic wave equation, the hyperbolic model for a chemosensitive movement, the nozzle flow and a two phase flow model. Properties of the scheme for the shallow water equations (Xing and Shu 2005, J. Comput. phys. 208, 206–227), such as the exact preservation of the balance laws for certain steady state solutions, the non-oscillatory property for general solutions with discontinuities, and the genuine high order accuracy in smooth regions, are maintained for the scheme when applied to this general class of hyperbolic systems     

An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements

The flux reconstruction approach offers an efficient route to high-order accuracy on unstructured grids. The location of the solution points plays an important role in determining the stability and accuracy of FR schemes on triangular elements. In particular, it is desirable that a solution point set (i) defines a well conditioned nodal basis for representing the solution, (ii) is symmetric, (iii) has a triangular number of points and, (iv) minimises aliasing errors when constructing a polynomial representation of the flux. In this paper we propose a methodology for generating solution points for triangular elements. Using this methodology several thousand point sets are generated and analysed. Numerical performance is assessed through an Euler vortex test case. It is found that the Lebesgue constant and quadrature strength of the points are strong indicators of stability and performance. Further, at polynomial orders \(\wp = 4,6,7\) solution points with superior performance to those tabulated in literature are discovered.     

A Robust Reconstruction for Unstructured WENO Schemes

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this paper, we combine two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. Numerical examples including both scalar and system cases are given to demonstrate stability and accuracy of the scheme.     

曲线的保面积细分算法

针对文献(Gordon D.Corner cutting and augmentation:An area-preserving method for smoothing polygonsand polylines.Computer Aided Geometric Design,2010,27(7):551-562)中给出的CCA1算法做了改进,提出了曲线保面积细分算法——CCA(k)细分算法.该算法将CCA1中的割角由等腰三角形推广至割角两边与特征多边形的相邻两边成比例,从而使极限曲线能更好地契合初始的特征多边形.文中还推导了CCA(k)算法的递推关系式和割比的适定取法,并证明了极限曲线的收敛性和连续性.数值实例表明,对于大多数的封闭多边形,CCA(k)算法都能得到理想的细分结果.     

Numerical Study of Compressible Mixing Layers Using High-Order WENO Schemes

This paper reports high resolution simulations using fifth-order weighted essentially non-oscillatory (WENO) schemes with a third-order TVD Runge-Kutta method to examine the features of turbulent mixing layers. The implementation of high-order WENO schemes for multi-species non-reacting Navier-Stokes (NS) solver has been validated through selective test problems. A comparative study of performance behavior of different WENO schemes has been made on a 2D spatially-evolving mixing layer interacting with oblique shock. The Bandwidth-optimized WENO scheme with total variation relative limiters is found to be less dissipative than the classical WENO scheme, but prone to exhibit some dispersion errors in relatively coarse meshes. Based on its accuracy and minimum dissipation error, the choice of this scheme has been made for the DNS studies of temporally-evolving mixing layers. The results are found in excellent agreement with the previous experimental and DNS data. The effect of density ratio is further investigated, reflecting earlier findings of the mixing growth-rate reduction.     

Stability of Some Generalized Godunov Schemes With Linear High-Order Reconstructions

Classical Semi-Lagrangian schemes have the advantage of allowing large time steps, but fail in general to be conservative. Trying to keep the advantages of both large time steps and conservative structure, Flux-Form Semi-Lagrangian schemes have been proposed for various problems, in a form which represent (at least in a single space dimension) a high-order, large time-step generalization of the Godunov scheme. At the theoretical level, a recent result has shown the equivalence of the two versions of Semi-Lagrangian schemes for constant coefficient advection equations, while, on the other hand, classical Semi-Lagrangian schemes have been proved to be strictly related to area-weighted Lagrange–Galerkin schemes for both constant and variable coefficient equations. We address in this paper a further issue in this theoretical framework, i.e., the relationship between stability of classical and of Flux-Form Semi-Lagrangian schemes. We show that the stability of the former class implies the stability of the latter, at least in the case of the one-dimensional linear continuity equation.     

一类多参数的曲线细分格式

构造了一类收敛的多参数差分格式,根据细分格式和差分格式的关系以及连续性条件可得到任意阶连续的多参数曲线细分格式.通过选取合适的参数可以得到一些经典的曲线细分格式,如Chaikin格式、三次样条细分格式和四点插值格式等;同时设计了一种C1连续的不对称三点插值格式,可以生成不对称的极限曲线.给出了同阶差分格式线性组合的性质,从而可设计出更多收敛的多参数曲线细分格式.     

一类可证安全的基于身份代理签名体制

当前,基于身份的数字签名体制研究取得丰硕的成果,利用双线性映射的GDH（Gap Diffie-Hellman）问题,人们提出了许多可证安全的基于身份密码体制。提供了一种直接利用现有的基于身份的数字签名体制,构造基于身份代理签名体制的途径,并利用可证安全理论,在标准计算模型下给出了严格的安全性证明。