A Stability Criterion for Semi-Discrete Difference Schemes of Hyperbolic Conservation Laws on Uniform Grids

In the present study, the stability condition for semi-discrete difference schemes of hyperbolic conservation laws obtained from Fourier analysis is simplified. This stability condition can be applied only to linear difference schemes with constant coefficients implemented with periodic boundary treatment. It could often give useful results for other cases, such as schemes with variable coefficients, schemes for nonperiodic problem and nonlinear problem. However, this condition usually leads to a trigonometric inequality, which makes it not convenient to use. For explicit difference schemes on uniform grids, this trigonometric inequality can be converted to polynomial form. Furthermore, if the scheme is a high-order one, the polynomial can be factorized into a simple form. Thus, it is much easier to solve than the inequality obtained directly from Fourier analysis. For compact difference schemes and conservative schemes, similar results are obtained. Some applications of this new stability criterion are shown, including judging the stability of two schemes, proving the upstream central schemes to be stable, constructing a stable upwind dissipation relation preserving (DRP) scheme and constructing an optimized weighted essentially non-oscillatory (WENO) scheme. Since WENO schemes are nonlinear schemes, the stability analysis in the present study is performed on their underlying linear schemes. According to the numerical tests, the underlying linear scheme should be stable, otherwise the corresponding WENO scheme may display instability. These applications demonstrate that this criterion is convenient and efficient for judging the linear stability of semi-discrete difference schemes and constructing stable upwind difference schemes.     

Parallel Iterative Schemes of Linear Algebra with Application to the Stability Analysis of Solutions of Systems of Linear Differential Equations

Parallel modifications of linear iteration schemes are proposed that are used to solve systems of liner algebraic equations and to achieve the time complexity equal to T = log₂ k · O(log₂ n), where k is the number of iterations of an original scheme and n is the dimension of a system. Such schemes are extended to the case of approximate solution of systems of linear differential equations with constant coefficients. Based on them and using a program, the stability of solutions in the Lyapunov sense is analyzed.     

Efficient Smoke Simulation on Curvilinear Grids

We present an efficient approach for performing smoke simulation on curvilinear grids. Our technique is based on a fast unconditionally‐stable advection algorithm and on a new and efficient solution to enforce mass conservation. It uses a staggered‐grid variable arrangement, and has linear cost on the number of grid cells. Our method naturally integrates itself with overlapping‐grid techniques, lending to an efficient way of producing highly‐realistic animations of dynamic scenes. Compared to approaches based on regular grids traditionally used in computer graphics, our method allows for better representation of boundary conditions, with just a small increment in computational cost. Thus, it can be used to evaluate aerodynamic properties, possibly enabling unexplored applications in computer graphics, such as interactive computation of lifting forces on complex objects. We demonstrate the effectiveness of our approach, both in 2‐D and 3‐D, through a variety of high‐quality smoke animations.     

A Numerical Strategy for Freestream Preservation of the High Order Weighted Essentially Non-oscillatory Schemes on Stationary Curvilinear Grids

The weighted essentially non-oscillatory (WENO) schemes have been extensively employed for the simulation of complex flow fields due to their high order accuracy and good shock-capturing properties. However, the standard finite difference WENO scheme cannot hold freestream automatically in general curvilinear coordinates. Numerical errors from non-preserved freestream can hide small scales such as turbulent flow structures; aero-acoustic waves which can make the results inaccurate or even cause the simulation failure. To address this issue, a new numerical strategy to ensure freestream preservation properties of the WENO schemes on stationary curvilinear grids is proposed in this paper. The essential idea of this approach is to offset the geometrically induced errors by proper discretization of the metric invariants. It includes the following procedures: (1) the metric invariants are retained in the governing equations and the full forms of the transformed equations on the general curvilinear coordinates are solved; (2) the symmetrical, conservative form of the metrics instead of the original ones are used; (3) the WENO schemes which are applied for the inviscid fluxes of the governing equations are employed to compute the outer-level partial derivatives of the metric invariants. In other words, the outer-level derivative operators for the metric invariants are kept the same with those for the corresponding inviscid fluxes. It is verified theoretically in this paper that by using this approach, the WENO schemes hold the freestream preservation properties naturally and thus work well in the generalized coordinate systems. For some well-known WENO schemes, the derivative operators for the metric invariants are explicitly expressed and thus this approach can be straightforwardly employed. The effectiveness of this strategy is validated by several benchmark test cases.     

Stability of Difference Splitting Schemes for Convection Diffusion Equation

We consider numerical modeling of the propagation of pollution in the air on the basis of geometrical splitting method for three-dimensional nonstationary convection diffusion equations. Splitting difference schemes in the form of explicit computing schemes are proposed to solve the obtained one-dimensional problems. The approximation, monotonicity, and stability of the proposed difference schemes are investigated.     

A Survey of User Interfaces for Computer Algebra Systems

This paper surveys work within the Computer Algebra community (and elsewhere) directed towards improving user interfaces for scientific computation during the period 1963–1994. It is intended to be useful to two groups of people: those who wish to know what work has been done and those who would like to do work in the field. It contains an extensive bibliography to assist readers in exploring the field in more depth. Work related to improving human interaction with computer algebra systems is the main focus of the paper. However, the paper includes additional materials on some closely related issues such as structured document editing, graphics, and communication protocols.     

Constructions in Finite Geometry Using Computer Algebra Systems

One way of using a computer algebra system to do research in finite geometry is to use the system to construct “small" order examples of various constructions, and then hope to recognize a pattern that can be generalized and eventually proven. Of course, initially one does not know if the “small" order examples exist. However, if one has sufficiently good insight concerning where to look and a reasonably good “starter", the computer algebra system will often find these examples quite expeditiously. Once found the system can then be used to analyze the constructs. Brute-force searching, on the other hand, is typically foolhardy with such general purpose systems. These ideas will be illustrated with two problems in finite geometry: (1) finding new translation planes by a technique called “nesting", and (2) finding large collections of pairwise disjoint projective bundles of conics.     

Vertex-Centered Linearity-Preserving Schemes for Nonlinear Parabolic Problems on Polygonal Grids

On arbitrary polygonal grids, a family of vertex-centered finite volume schemes are suggested for the numerical solution of the strongly nonlinear parabolic equations arising in radiation hydrodynamics and magnetohydrodynamics. We define the primary unknowns at the cell vertices and derive the schemes along the linearity-preserving approach. Since we adopt the same cell-centered diffusion coefficients as those in most existing finite volume schemes, it is required to introduce some auxiliary unknowns at the cell centers in the case of nonlinear diffusion coefficients. A second-order positivity-preserving algorithm is then suggested to interpolate these auxiliary unknowns via the primary ones. All the schemes lead to symmetric and positive definite linear systems and their stability can be rigorously analyzed under some standard and weak geometry assumptions. More interesting is that these vertex-centered schemes do not have the so-called numerical heat-barrier issue suffered by many existing cell-centered or hybrid schemes (Lipnikov et al. in J Comput Phys 305:111–126, 2016). Numerical experiments are also presented to show the efficiency and robustness of the schemes in simulating nonlinear parabolic problems.     

On the Asymptotic Stability Problem for Solutions of Difference Switched Systems

We consider difference systems obtained by discretizing certain classes of differential systems. It is assumed that the system under consideration can operate in several modes. The problem is to establish conditions that guarantee the asymptotic stability of a given equilibrium position when switching regimes. We use the method of Lyapunov functions. We study the case when solutions of the system under various operating modes can have features of both linear and nonlinear behavior.     

非线性系统观测器的计算机代数分析与设计方法*

本文介绍了控制系统非线性观测器的分析和设计方法，根据扩展Ｌｕｅｎｂｅｒｇｅｒ观测器设计过程中放大系数向量的计算公式，用计算机代数方法进行非线性观测器设计，本文给出了用Ｍａｔｈｅｍａｔｉｃａ符号编程语言实现的算法软件包，并使用该软件包对一个具体实例进行了分析和设计。     

计算机控制系统的故障分析

1　引言随着当前自动控制系统规模的增大 ,结构的日趋复杂 ,越来越多的控制系统中引入了计算机 ,由计算机来直接控制系统的工作状况。此时 ,整个控制系统的安全性、可靠性、准确性问题自然就落在了计算机上。本文主要介绍控制系统中计算机所面临的各种故障以及解决方案。衡量计算机控制系统可靠性的一个重要参数是可利用率 ,即A =t MTBF/(t MTBF t MTTR)式中 ,A是指系统正常工作时间与总时间的比率 ,t MTTR是平均维修时间 ,即可修复系统从故障至恢复正常工作所利用的修复时间的平均值。当延长 t MTBF,缩短 t MTTR 才能提高系统…     

利用流SIMD扩展加速3D曲线网格的流线计算

流线是一种基本的流场可视化技术,计算流线要耗费大量时间,Intel处理器（Pentium Ⅲ,Pentium4）提供流SIMD扩展（SSE）,支持指令 级SIMD操作。3D曲线网格上的流线计算包含速度插值、数值积分、点定位等主要子过程,具有很高的内在SIMD并行性。通过将数据按SSE数据类型组织以及对主要子过程进行SIMD并行化,设计了线流计算的SSE算法。采用向量类库、嵌入汇编两种SSE编码方式分别实现SSE算法,并依据处理器的体系结构优化代码。测试结果表明：SSE大大加速了3D曲线网格的流线计算,向量类库方式比传统计算提高55％左右的性能,嵌入汇编提高75％左右。     

Combining SAT Solvers with Computer Algebra Systems to Verify Combinatorial Conjectures

We present a method and an associated system, called MathCheck, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, à la MathCheck, can be used as an assistant by mathematicians to either find counterexamples or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL (T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the capabilities of several different CAS, namely the SAGE, Maple, and Magma systems. As case studies, we study three long-standing open mathematical conjectures, two from graph theory regarding properties of hypercubes, and one from combinatorics about Hadamard matrices. The first conjecture states that any matching of any d-dimensional hypercube can be extended to a Hamiltonian cycle; the second states that given an edge-antipodal coloring of a hypercube there always exists a monochromatic path between two antipodal vertices; the third states that Hadamard matrices exist for all orders divisible by 4. Previous results on the graph theory conjectures have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. Regarding Hadamard matrices, we demonstrate the advantages of SAT+CAS by constructing Williamson matrices up to order 42 (equivalently, Hadamard up to order \(4\times 42=168\)), improving the bounds up to which Williamson matrices of even order have been constructed. Prior state-of-the-art construction was only feasibly performed for odd numbers, where possible.     

Components for Algebra Transformation Systems

In this paper we study the instantiation of the generic notion of component introduced before in terms of algebra transformation systems, where refinements are defined by means of high-level replacement rules. With this work we follow two aims. On one hand we show that our generic component concept is not purely syntactic, but it may take into account the semantics of the specifications involved. On the other hand, with this instantiation we strengthen the role of our component concept in the study of component-based heterogeneous systems.     

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     

基于模型的网络控制系统稳定性分析及应用

针对时延网络控制系统中被控对象状态无法直接测量的情况,提出了基于模型的时延网络控制系统模型。系统采用状态反馈控制器,在网络信号传输时间间隔内,以被控对象模型为依据,计算控制信号,以减少系统对网络的依赖。在此基础上,给出了网络控制系统全局指数稳定的充要条件,该条件受网络信号更新时间、被控对象模型误差及网络引起的时延等因素的影响。仿真示例验证了该条件的有效性,应用在工业控制系统中,取得了较好的控制效果。