Explicit construction of an autonomous Hamiltonian system exhibiting continual directed flow

We construct a prototypical example of a spatially-open autonomous Hamiltonian system in which localised, but otherwise unbiased, ensembles of initial conditions break spatio-temporal symmetries in the subsequent ensemble dynamics, despite time reversal symmetry of the equations of motion. Together with transient chaos, this provides the mechanism for the occurrence of a current. Transporting trajectories pass through transient chaos and subsequently cross surfaces of no-return, after which they perform solely regular motion so that the current is of continual ballistic nature.     

A high-order energy-conserving integration scheme for Hamiltonian systems

A new energy-conserving numerical integration method for Hamiltonian systems is presented. The method is constructed by a parallel connection of n multi-stage schemes of order 2 and its order of accuracy is 2n.     

A central conservative scheme for general rectangular grids

We present an extension of the genuinely multi-dimensional semi-discrete central scheme developed in [A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740.] to arbitrary orthogonal grids. The presented algorithm is constructed to yield the geometric scaling factors in a self-consistent way.Additionally, the order of the scheme is not fixed during the derivation of the basic algorithm. Based on the resulting general scheme it is possible to construct methods of any desired order, just by considering the corresponding reconstruction polynomial. We demonstrate how a second order scheme in plane polar coordinates and cylindrical coordinates can be derived from our general formulation. Finally, we demonstrate the correctness of this second order scheme through application to several numerical experiments.     

A conservative phase field method for solving incompressible two-phase flows

In this paper a conservative phase-field method based on the work of Sun and Beckermann [Y. Sun, C. Beckermann, Sharp interface tracking using the phase-field equation, J. Comput. Phys. 220 (2007) 626–653] for solving the two- and three-dimensional two-phase incompressible Navier–Stokes equations is proposed. The present method can preserve the total mass as the Cahn–Hilliard equation, but the calculation and implementation are much simpler than that. The dispersion-relation-preserving schemes are utilized for the advection terms while the Helmholtz smoother is applied to compute the surface-tension force term. To verify the proposed method, several benchmarks are examined and shown to have good agreements with previous results. It also shows that the satisfactions of mass conservations are guaranteed.     

A Hermite upwind WENO scheme for solving hyperbolic conservation laws

This paper proposes a new WENO procedure to compute problems containing both discontinuities and a large disparity of characteristic scales.In a one-dimensional context, the WENO procedure is defined on a three-points stencil and designed to be sixth-order in regions of smoothness. We define a finite-volume discretization in which we consider the cell averages of the variable and its first derivative as discrete unknowns. The reconstruction of their point-values is then ensured by a unique sixth-order Hermite polynomial. This polynomial is considered as a symmetric and convex combination, by ideal weights, of three fourth-order polynomials: a central polynomial, defined on the three-points stencil, is combined with two polynomials based on the left and the right two-points stencils.The symmetric nature of such an interpolation has an important consequence: the choice of ideal weights has no influence on the properties of the discretization. This advantage enables to formulate the Hermite interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, nonlinear weights are then defined.To deal with the peculiarities of the Hermite interpolation near discontinuities, we define a new procedure in order for the nonlinear weights to smoothly evolve between the ideal weights, in regions of smoothness, and one-sided weights, otherwise.The resulting scheme is a sixth-order WENO method based on central Hermite interpolation and TVD Runge–Kutta time-integration. We call this scheme the HCWENO6 scheme.Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In these experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems.     

Guided random walks for solving Hamiltonian lattice gauge theories

Motivated by developments for many-particle quantum systems, a Monte Carlo method for solving Hamiltonian lattice gauge theories without fermions is presented in which a stochastic random walk is guided by a trial wave function. To the extent that a substantial portion of the local structure of the theory can be incorporated in the trial function, the method offers significant advantages relative to existing techniques. The method is applicable to the study of SU(N) lattice gauge theories, and its utility is demonstrated by solving the compact U(1) gauge theory in three spatial dimensions.     

Simple approximation scheme for the Anderson impurity Hamiltonian

A simple approximation scheme is presented for solving the Anderson impurity model within the Noncrossing Approximation (NCA). It is shown that with it the computations reduce to an extent that the scheme can be readily applied to interpret different experiments. The theory is applied to calculate the temperature dependence of the quadrupole moment and of the static and dynamic magnetic susceptibility. The effects of the crystalline electric field (CEF) are thereby incorporated. A comparison of the static susceptibility with exact Bethe ansatz results is given, when the effect of the CEF is neglected. Comparisons with the numerically exact solutions of the NCA are made where they are available. The application of the present theory to YbCu₂Si₂ is discussed.     

Canonical methods for Hamiltonian systems: numerical experiments

A Hamiltonian system possesses dynamics (e.g. preservation of volume in phase space and symplectic structure) that call for special numerical integrators, namely canonical methods. Recent research on this aspect have shown that canonical numerical integrators may be needed for Hamiltonian systems. In this paper, we focus on numerical experiments that compare canonical and non-canonical numerical integrators. Test problems are taken from different areas in physical sciences. These experiments help to buttress the claims that canonical numerical integrators give results that mimic the qualitative behavior of the original system and that canonical numerical integrators are suitable for long time integrations. Our experiments indicate that higher-order canonical methods allow for larger timestep than lower-order canonical methods.     

Characteristic analysis of 5D symmetric Hamiltonian conservative hyperchaotic system with hidden multiple stability

Conservative chaotic systems have unique advantages over dissipative chaotic systems in the fields of secure communication and pseudo-random number generator because they do not have attractors but possess good traversal and pseudorandomness. In this work, a novel five-dimensional(5D) Hamiltonian conservative hyperchaotic system is proposed based on the 5D Euler equation. The proposed system can have different types of coordinate transformations and time reversal symmetries. In this work, Hamilt...     

A second order numerical scheme for the improved Boussinesq equation

A finite-difference scheme arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation is used for the numerical solution of the improved Boussinesq equation (IBq). The resulting linear scheme, which is analyzed for local truncation error and stability, is tested numerically and conclusions with corresponding results known in the bibliography are derived.     

A high order ENO conservative Lagrangian type scheme for the compressible Euler equations

We develop a class of Lagrangian type schemes for solving the Euler equations of compressible gas dynamics both in the Cartesian and in the cylindrical coordinates. The schemes are based on high order essentially non-oscillatory (ENO) reconstruction. They are conservative for the density, momentum and total energy, can maintain formal high order accuracy both in space and time and can achieve at least uniformly second-order accuracy with moving and distorted Lagrangian meshes, are essentially non-oscillatory, and have no parameters to be tuned for individual test cases. One and two-dimensional numerical examples in the Cartesian and cylindrical coordinates are presented to demonstrate the performance of the schemes in terms of accuracy, resolution for discontinuities, and non-oscillatory properties.     

A high order ENO conservative Lagrangian type scheme for the compressible Euler equations

We develop a class of Lagrangian type schemes for solving the Euler equations of compressible gas dynamics both in the Cartesian and in the cylindrical coordinates. The schemes are based on high order essentially non-oscillatory (ENO) reconstruction. They are conservative for the density, momentum and total energy, can maintain formal high order accuracy both in space and time and can achieve at least uniformly second-order accuracy with moving and distorted Lagrangian meshes, are essentially non-oscillatory, and have no parameters to be tuned for individual test cases. One and two-dimensional numerical examples in the Cartesian and cylindrical coordinates are presented to demonstrate the performance of the schemes in terms of accuracy, resolution for discontinuities, and non-oscillatory properties.     

A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes

This paper proposes a new WENO procedure to compute multi-scale problems with embedded discontinuities, on non-uniform meshes.In a one-dimensional context, the WENO procedure is first defined on a five-points stencil and designed to be fifth-order accurate in regions of smoothness. To this end, we define a finite-volume discretization in which we consider the cell averages of the variable as the discrete unknowns. The reconstruction of their point-values is then ensured by a unique fifth-order polynomial. This optimum polynomial is considered as a symmetric and convex combination, by ideal weights, of four quadratic polynomials.The symmetric nature of the resulting interpolation has an important consequence: the choice of ideal weights has no influence on the accuracy of the discretization. This advantage enables to formulate the interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights.We adapt this procedure for the non-linear weights to maintain the theoretical convergence properties of the optimum reconstruction, whatever the problem considered.The resulting scheme is a fifth-order WENO method based on central interpolation and TVD Runge–Kutta time-integration. We call this scheme the CWENO5 scheme.Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In those experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems. Finally, the new algorithm is directly extended to bi-dimensional problems.     

一种新的三维自治混沌系统

通过对蔡氏电路的研究,提出了一种新的混沌系统,并对该系统的基本动力学特性进行了深入研究,得到该系统的Lyapunov指数和Lyapunov维数,给出了相图、Lyapunov指数谱、分岔图、Poincaré映射以及功率谱等.利用OrCAD-PSpice软件设计了该新混沌系统的振荡电路并进行了仿真实验.研究结果表明,该系统与蔡氏电路产生的混沌吸引子并不拓扑等价,且该系统的参数变化范围较大,最大Lyapunov指数接近1,数值仿真和电路系统实验仿真具有很好的一致性,证实了该系统的存在性和物理上可实现性. 关键词： 混沌系统 Lyapunov指数谱 分岔图 电路实现     

A novel four-dimensional autonomous hyperchaotic system

A novel four-dimensional autonomous hyperchaotic system is reported in this paper. Some basic dynamical properties of the new hyperchaotic system are investigated in detail by means of a continuous spectrum, Lyapunov exponents, fractional dimensions, a strange attractor and Poincaré mapping. The dynamical behaviours of the new hyperchaotic system are proved by not only performing numerical simulation and brief theoretical analysis but also by conducting an electronic circuit experiment.     

A numerical scheme for particle-laden thin film flow in two dimensions

The physics of particle-laden thin film flow is not fully understood, and recent experiments have raised questions with current theory. There is a need for fully two-dimensional simulations to compare with experimental data. To this end, a numerical scheme is presented for a lubrication model derived for particle-laden thin film flow in two dimensions with surface tension. The scheme relies on an ADI process to handle the higher-order terms, and an iterative procedure to improve the solution at each timestep. This is the first paper to simulate the two-dimensional particle-laden thin film lubrication model. Several aspects of the scheme are examined for a test problem, such as the timestep, runtime, and number of iterations. The results from the simulation are compared to experimental data. The simulation shows good qualitative agreement. It also suggests further lines of inquiry for the physical model.     

A local energy-preserving scheme for Zakharov system

In this paper, we propose a local conservation law for the Zakharov system. The property is held in any local timespace region which is independent of the boundary condition and more essential than the global energy conservation law.Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose a local energy-preserving(LEP) scheme for the system. The merit of the proposed scheme is that the local energy conservation law can be conserved exactly in any time-space region. With homogeneous Dirchlet boundary conditions, the proposed LEP scheme also possesses the discrete global mass and energy conservation laws. The theoretical properties are verified by numerical results.