Data mining techniques for scientific computing: Application to asymptotic paraxial approximations to model ultrarelativistic particles

We propose a new approach that consists in using data mining techniques for scientific computing. Indeed, data mining has proved to be efficient in other contexts which deal with huge data like in biology, medicine, marketing, advertising and communications. Our aim, here, is to deal with the important problem of the exploitation of the results produced by any numerical method. Indeed, more and more data are created today by numerical simulations. Thus, it seems necessary to look at efficient tools to analyze them. In this work, we focus our presentation to a test case dedicated to an asymptotic paraxial approximation to model ultrarelativistic particles. Our method directly deals with numerical results of simulations and try to understand what each order of the asymptotic expansion brings to the simulation results over what could be obtained by other lower-order or less accurate means. This new heuristic approach offers new potential applications to treat numerical solutions to mathematical models.     

Dynamics of rotating two-component Bose-Einstein condensates and its efficient computation

In this paper, we investigate the dynamics of rotating two-component Bose-Einstein condensates (BEC) based on the coupled Gross-Pitaevskii equations (CGPEs) with an angular momentum rotation term and an external driving field, and propose an efficient and accurate method for numerical simulations. We prove the conservation of the angular momentum expectation, derive the dynamic laws for the density of each component and condensate widths, and analyze the dynamics of a stationary state with its center shifted from the trap center. By formulating the CGPEs in either 2D (two-dimensional) polar coordinate or 3D cylindrical coordinate system, the angular momentum rotation term becomes a term with constant coefficients. This allows us to develop an efficient time-splitting method which is time reversible, time transverse invariant, unconditionally stable, efficient and accurate for the problem. Moreover, it conserves the total position density in the discretized level. The numerical method is applied to verify our analytical results and study the dynamics of quantized vortex lattices in rotating two-component BEC with/without an external driving field.     

一个新的三维二次混沌系统及其电路实现

为了产生复杂的混沌吸引子，构造了一个新的三维二次自治混沌系统.该系统含有三个参数，每一个方程含有一个非线性乘积项.利用理论推导、数值仿真、Lyapunov指数谱和分岔图对系统的基本动力学特性进行了分析.结果表明，该系统具有五个平衡点，因而与Lorenz，Rsslor，Chen、Lü等混沌系统是非拓扑等价的；当其参数满足一定条件时，系统是混沌的.与Lorenz等混沌系统相比，该系统具有更大的正Lyapunov指数，能够产生复杂的混沌吸引子和一些有趣的动力学行为.最后，设计了实现该系统的混沌电路，电路实验结 关键词： 三维二次自治系统 混沌 混沌吸引子 电路实现     

First- and second-order finite volume methods for the one-dimensional nonconservative Euler system

Gas flow in porous media with a nonconstant porosity function provides a nonconservative Euler system. We propose a new class of schemes for such a system for the one-dimensional situations based on nonconservative fluxes preserving the steady-state solutions. We derive a second-order scheme using a splitting of the porosity function into a discontinuous and a regular part where the regular part is treated as a source term while the discontinuous part is treated with the nonconservative fluxes. We then present a classification of all the configurations for the Riemann problem solutions. In particularly, we carefully study the resonant situations when two eigenvalues are superposed. Based on the classification, we deal with the inverse Riemann problem and present algorithms to compute the exact solutions. We finally propose new Sod problems to test the schemes for the resonant situations where numerical simulations are performed to compare with the theoretical solutions. The schemes accuracy (first- and second-order) is evaluated comparing with a nontrivial steady-state solution with the numerical approximation and convergence curves are established.     

Relativistic Ornstein–Uhlenbeck Process

We wish to shed some light on the problem of thermodynamic irreversibility in the relativistic framework. Therefore, we propose a relativistic stochastic process based on a generalization of the usual Ornstein–Uhlenbeck process: we introduce a relativistic version of the Langevin equation with a damping term which has the correct Galilean limit. We then deduce relativistic Kramers and Fokker–Planck equations and a fluctuation-dissipation theorem is derived from them. Finally, numerical simulations are used to check the equilibrium distribution in momentum space and to investigate diffusion in physical space.     

The effect of diffusion in a new viscous continuum traffic model

In this Letter, we propose a new continuum traffic model with a viscous term. The linear stability condition for viscous shock waves is derived. We derive the Korteweg-de Vries (KdV) equation near the neutral stability line. Then we investigate the effect of the viscous term by numerical simulations. The results show that viscosity may induce oscillations and the amplitude of the oscillation increases as the viscosity coefficient increases. This agrees with the linear stability condition. The local clusters are compressed by increasing the viscosity coefficient in the cluster study.     

Numerical simulation of a solitonic gas in KdV and KdV–BBM equations

The collective behaviour of soliton ensembles (i.e. the solitonic gas) is studied using the methods of the direct numerical simulation. Traditionally this problem was addressed in the context of integrable models such as the celebrated KdV equation. We extend this analysis to non-integrable KdV–BBM type models. Some high resolution numerical results are presented in both integrable and nonintegrable cases. Moreover, the free surface elevation probability distribution is shown to be quasi-stationary. Finally, we employ the asymptotic methods along with the Monte Carlo simulations in order to study quantitatively the dependence of some important statistical characteristics (such as the kurtosis and skewness) on the Stokes–Ursell number (which measures the relative importance of nonlinear effects compared to the dispersion) and also on the magnitude of the BBM term.     

基于无导数优化方法的数值模式误差估计

初始场误差和模式误差是制约数值预报准确率的两个关键因素,本文主要考虑利用历史观测资料实现时空演变的模式误差的估计问题.通过把模式误差综合考虑成为准确模式中的未知项,把历史资料看作是带有未知项的准确模式的特解,构造了求解时空演变的模式误差项的反问题及其最优控制问题.给出了一个解决最优控制问题的无导数优化方法,该方法的优点是不需要建立原数值模式的切线性模式与伴随模式,它只需在增加一个外强迫项的基础上运行原数值模式即可实现模式误差项的最优估计.关于Burgers方程的算例表明,无论模式的初始状态是否准确已知,无导数优化方法都能有效解决时空演变的模式误差的最优估计问题,它为实际业务模式利用历史数据提取模式误差信息并显著地改进预报效果提供了一种方便可行的数值方法与理论依据.     

Near-field performance analysis of locally-conformal perfectly matched absorbers via Monte Carlo simulations

In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of our new PML approach, which we call as locally-conformal PML, using Monte Carlo simulations. The locally-conformal PML method is an easily implementable conformal PML implementation, to the problem of mesh truncation in the FEM. The most distinguished feature of the method is its simplicity and flexibility to design conformal PMLs over challenging geometries, especially those with curvature discontinuities, in a straightforward way without using artificial absorbers. The method is based on a special complex coordinate transformation which is ‘locally-defined’ for each point inside the PML region. The method can be implemented in an existing FEM software by just replacing the nodal coordinates inside the PML region by their complex counterparts obtained via complex coordinate transformation. We first introduce the analytical derivation of the locally-conformal PML method for the FEM solution of the two-dimensional scalar Helmholtz equation arising in the mathematical modeling of various steady-state (or, time-harmonic) wave phenomena. Then, we carry out its numerical performance analysis by means of some Monte Carlo simulations which consider both the problem of constructing the two-dimensional Green’s function, and some specific cases of electromagnetic scattering.     

Near-field performance analysis of locally-conformal perfectly matched absorbers via Monte Carlo simulations

In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of our new PML approach, which we call as locally-conformal PML, using Monte Carlo simulations. The locally-conformal PML method is an easily implementable conformal PML implementation, to the problem of mesh truncation in the FEM. The most distinguished feature of the method is its simplicity and flexibility to design conformal PMLs over challenging geometries, especially those with curvature discontinuities, in a straightforward way without using artificial absorbers. The method is based on a special complex coordinate transformation which is ‘locally-defined’ for each point inside the PML region. The method can be implemented in an existing FEM software by just replacing the nodal coordinates inside the PML region by their complex counterparts obtained via complex coordinate transformation. We first introduce the analytical derivation of the locally-conformal PML method for the FEM solution of the two-dimensional scalar Helmholtz equation arising in the mathematical modeling of various steady-state (or, time-harmonic) wave phenomena. Then, we carry out its numerical performance analysis by means of some Monte Carlo simulations which consider both the problem of constructing the two-dimensional Green’s function, and some specific cases of electromagnetic scattering.     

Applications of a Cole–Hopf transform to the 3D Navier–Stokes equations

The Navier–Stokes equations written in the vector potential can be recast as non-linear Schrödinger equations at imaginary times, i.e. heat equations with a potential term, using the Cole–Hopf transform. On this basis, we study two kinds of Navier–Stokes flows by means of direct numerical simulations. In an experiment on vortex reconnection, it is found that the potential term takes large negative values in regions where intensive reconnection takes place, whereas the signature of the non-linear term is more broadly spread. For decaying turbulence starting from a random initial condition, such a correspondence is also observed in the early stage when the flow is dominated by vorticity layers. At later times, when the flow features several tubular vortices, this correspondence becomes weaker. Finally, a similar set of transformations is presented for the magneto–hydrodynamic equations, which reduces them to a set of heat equations with suitable potential terms, thereby obtaining new criteria for the regularity of their solutions.     

Heat Flux for a Relativistic Dilute Bidimensional Gas

Relativistic kinetic theory predicts substantial modifications to the dissipation mechanisms of a dilute gas. For the heat flux, these include (in the absence of external forces) a correction to the thermal conductivity and the appearance of a new, purely relativistic, term proportional to the density gradient. In this work we obtain such constitutive equation for the particular case of a bidimensional gas. The calculation is based on the Chapman–Enskog solution to the relativistic Boltzmann equation and yields analytical expressions for the corresponding transport coefficients, which are evaluated for the particular case of hard disks. These results will be useful for numerical simulations and may be applied to bidimensional non-dense materials.     

具有完全不确定参数的五项双曲型混沌系统的投影同步

提出了一个新的简单的双曲型三维自治混沌系统,该三维混沌系统只含有五项, 并且其非线性特征主要依赖于一个非线性二次双曲正弦项和一个非线性二次交叉项. 较已有的三维混沌系统而言, 不仅系统的项要少一些, 而且在参数变化时, 呈现混沌的参数范围也很大. 对系统的一些基本动力学特性进行了数值模拟和理论分析. 同时, 还研究了具有完全不确定参数的该五项双曲型混沌系统的投影同步. 基于Lyapunov指数稳定性理论和Barbalat引理, 设计了一个新的具有参数自适应律的自适应同步控制器, 利用该控制器分别实现了两个结构相同和相异混沌系统的渐进性和全局性投影同步. 数值模拟验证了该方法的有效性和可行性.     

An ignition-temperature model with two free interfaces in premixed flames†

In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numerical analysis. We prove the existence of a critical Lewis number such that the travelling wave solution is stable for values of Lewis number below the critical one and is unstable for Lewis numbers that exceed this critical value. Finally, we discuss the results of numerical simulations of a fully nonlinear system that describes the perturbation dynamics of planar fronts. These simulations reveal, in particular, some very interesting ‘two-cell’ steady patterns of curved combustion fronts.     

A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation

To develop and analyze new computational techniques for the Boltzmann equation based on model or approximation adaptivity, it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the Boltzmann equation and that admits the extraction of highly accurate reference solutions. For standard collision processes, the Boltzmann equation itself fails to meet the second requirement for d = 2, 3 spatial dimensions, on account of its setting in 2d, while for d = 1 the first requirement is violated because the Boltzmann equation then lacks the convergence-to-equilibrium property that forms the substructure of simplified models. In this article we present a numerical investigation of a new one-dimensional prototype of the Boltzmann equation. The underlying molecular model is endowed with random collisions, which have been fabricated such that the corresponding Boltzmann equation exhibits convergence to Maxwell–Boltzmann equilibrium solutions. The new Boltzmann model is approximated by means of a discontinuous Galerkin (DG) finite-element method. To validate the one-dimensional Boltzmann model, we conduct numerical experiments and compare the results with Monte-Carlo simulations of equivalent molecular-dynamics models.     

A lattice Bhatnagar--Gross--Krook model for a class of the generalized Burgers equations

A new lattice Bhatnagar--Gross--Krook (LBGK) model for a class of the generalized Burgers equations is proposed. It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods. Satisfactory results are obtained by the numerical simulations.     

Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator

In the dynamics of optical systems, one commonly needs to cope with the problem of coexisting deterministic and stochastic components. The separation of these components is an important, although difficult, task. Often the time scales at which determinism and noise dominate the system's dynamics differ. In this Letter we propose to use information-theory-derived quantifiers, more precisely, permutation entropy and statistical complexity, to distinguish between the two behaviors. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrate that the time scales at which deterministic or noisy behavior dominate can be identified. Supporting numerical simulations prove the accuracy of this identification.     

美式回望期权定价问题的有限体积法

随着金融市场的不断发展, 期权作为一种能够规避风险的金融衍生产品越来越引起投资者的青睐, 成交量呈逐年上升的趋势, 期权定价问题已经成为金融数学领域中一个重要的研究课题. 本文主要研究Black-Scholes模型下美式回望期权定价问题的数值解法. 美式回望期权定价问题是一个二维非线性抛物问题, 难以直接应用数值方法进行求解. 通过分析该问题的求解难点, 本文给出解决该困难的有效方法. 首先利用计价单位变换将定价问题转换为一维自由边值问题, 并采用Landau's变换将求解区域规范化; 而后针对问题的非线性特点,利用有限体积法和Newton法交替迭代求解期权价格和最佳实施边界, 并对数值解的非负性进行了分析. 最后, 通过与二叉树方法进行比较, 验证了本文方法的正确性和有效性, 为实际应用提供了理论基础.     

Synchronization in an array of linearly stochastically coupled networks with time delays

In this paper, the complete synchronization problem is investigated in an array of linearly stochastically coupled identical networks with time delays. The stochastic coupling term, which can reflect a more realistic dynamical behavior of coupled systems in practice, is introduced to model a coupled system, and the influence from the stochastic noises on the array of coupled delayed neural networks is studied thoroughly. Based on a simple adaptive feedback control scheme and some stochastic analysis techniques, several sufficient conditions are developed to guarantee the synchronization in an array of linearly stochastically coupled neural networks with time delays. Finally, an illustrate example with numerical simulations is exploited to show the effectiveness of the theoretical results.     

Application of statistical mechanics to combinatorial optimization problems: The chromatic number problem andq-partitioning of a graph

Methods of statistical mechanics are applied to two important NP-complete combinatorial optimization problems. The first is the chromatic number problem, which seeks the minimal number of colors necessary to color a graph such that no two sites connected by an edge have the same color. The second is partitioning of a graph intoq equal subgraphs so as to minimize intersubgraph connections. Both models are mapped into a frustrated Potts model, which is related to theq- state Potts spin glass. For the first problem, we obtain very good agreement with numerical simulations and theoretical bounds using the annealed approximation. The quenched model is also discussed. For the second problem we obtain analytic and numerical results by evaluating the groundstate energy of theq=3 and 4 Potts spin glass using Parisi's replica symmetry breaking. We also perform some numerical simulations to test the theoretical result and obtain very good agreement.