Scaling,propagation, and kinetic roughening of flame fronts in random media

We introduce a model of two coupled reaction-diffusion equations to describe the dynamics and propagation of flame fronts in random media. The model incorporates heat diffusion, its dissipation, and its production through coupling to the background reactant density. We first show analytically and numerically that there is a finite critical value of the background density below which the front associated with the temperature field stops propagating. The critical exponents associated with this transition are shown to be consistent with meanfield theory of percolation. Second, we study the kinetic roughening associated with a moving planar flame front above the critical density. By numerically calculating the time-dependent width and equal-time height correlation function of the front, we demonstrate that the roughening process belongs to the universality class of the Kardar-Parisi-Zhang interface equation. Finally, we show how this interface equation can be analytically derived from our model in the limit of almost uniform background density.     

磁化尘埃等离子体中的冲击波及其横向扰动稳定性

利用双曲函数法得到ZKB方程的一组冲击波解,并对波在横向扰动下的动力学稳定性进行研究.对冲击波解进行线性稳定性分析,并构造高精度的有限差分格式求解所得本征值问题.结果表明：对于正耗散的情形,该冲击波在线性意义下稳定;对于负耗散情形,该冲击波在线性意义下不稳定.构造有限差分格式对受扰动的冲击波进行非线性动力学演化,结果表明：对于正耗散的情况,该冲击波是稳定的.     

Spin current pumping with two orthogonal electric fields in quantum wire

We study the problem of spin current pumping in a one-dimensional quantum wire when there exist two orthogonal Rashba spin-orbit couplings (SOCs) in different regions which evolve with time and can be induced by the perpendicular electric fields. On one hand, we demonstrate that the time-evolving Rashba SOC is equivalent to the spin-dependent electric field and the scheme may lead to the pure spin current associated with well suppressed charge current. On the other hand, we adopt the non-equilibrium Green's function method and numerically find that the parameter loop must satisfy certain condition for the successful pumping. We also study the effect of the Fermi energy and the inevitable disorder on the spin current. The implications of these results are discussed.     

非线性基底势对离散非线性晶格孤波动力学性质的影响

使用一种简化的准离散多重尺度法,研究了具有非线性基底势的一维离散非线性晶格的孤波解, 表明孤波能够在这种一维非线性晶格链中存在,而且非线性基底势对孤波的载波频率、群速度、振幅等动力学性质都将产生影响。 另一方面,我们还对非线性动力学方程进行了数值求解, 发现用简化的准离散多重尺度法得到的近似解与精确的计算机数值计算结果符合得较好。     

Hybrid Flux-Splitting Schemes for a Two-Phase Flow Model

In this paper we deal with the construction of hybrid flux-vector-splitting (FVS) schemes and flux-difference-splitting (FDS) schemes for a two-phase model for one-dimensional flow. The model consists of two mass conservation equations (one for each phase) and a common momentum equation. The complexity of this model, as far as numerical computation is concerned, is related to the fact that the flux cannot be expressed in terms of its conservative variables. This is the motivation for studying numerical schemes which are not based on (approximate) Riemann solvers and/or calculations of Jacobian matrix. This work concerns the extension of an FVS type scheme, a Van Leer type scheme, and an advection upstream splitting method (AUSM) type scheme to the current two-phase model. Our schemes are obtained through natural extensions of corresponding schemes studied by Y. Wada and M.-S. Liou (1997, SIAM J. Sci. Comput.18, 633–657) for Euler equations. We explore the various schemes for flow cases which involve both fast and slow transients. In particular, we demonstrate that the FVS scheme is able to capture fast-propagating acoustic waves in a monotone way, while it introduces an excessive numerical dissipation at volume fraction contact (steady and moving) discontinuities. On the other hand, the AUSM scheme gives accurate resolution of contact discontinuities but produces oscillatory approximations of acoustic waves. This motivates us to propose other hybrid FVS/FDS schemes obtained by removing numerical dissipation at contact discontinuities in the FVS and Van Leer schemes.     

Exciting rogue waves,breathers, and solitons in coherent atomic media

We propose a scheme that excites rogue waves via electromagnetically induced transparency(EIT), which can also excite breathers and solitons. The system is a resonant Λ-type atomic ensemble. Under EIT conditions, the envelope equation of the probe field can be reduced to several different models, such as the saturable nonlinear Schr?dinger equation(SNLSE), and SNLSE with the trapping potential provided by a far-detuned laser field or a magnetic field. In this scheme, rogue waves can be generated by different initial pulses, such as the Gaussian wave with(or without) the uniform background. The scheme can be used to obtain rogue waves,breathers and solitons. We show the existence regions of rogue waves, breathers, and solitons as the function of the amplitude and width of the initial pulse. The novelty of our paper is that, we not only show rogue waves in the integrable system numerically, but also present the method to generate and control rogue waves in the non-integrable system.     

K(m,n,P)方程多-Compacton相互作用的数值研究

采用有限差分法对非线性色散K(m,n,p)方程的多-Compacton之间的相互作用进行了数值研究.该差分方法为二阶精度且线性意义下绝对稳定的无耗散格式,通过添加人工耗散项有效防止了数值解的爆破现象.首先对单-Compacton的长时间演化行为进行了数值模拟,验证了数值方法的有效性.然后对双-CompaCton和三-Compacton的碰撞过程进行了数值研究,发现多-Compacton碰撞之后基本保持碰撞之前的波形和波速,但在波后产生小振幅的Compacton-Anticompacton对.     

Nonlinear instability of charged liquid jets: Effect of interfacial charge relaxation

A linear and nonlinear study has been made of cylindrical interface, carrying a uniform surface charge in the presence of a finite rate of charge relaxation, is investigated by using multiple scales method. The linear stability flow is analyzed by deriving a dispersion relation for the growth waves, and solving it analytically and numerically to find marginal stability curves. We investigate the electric charge relaxation effects on the stability of the flow by considering various limiting cases. We also examine the effects of finite charge relaxation times in axisymmetric and nonaxisymmetric modes. In the nonlinear approach, it is shown that the evolution of the amplitude is governed by a Ginzburg–Landau equation. There is also obtained a nonlinear modified Schrödinger equation describing the evolution of wave packets for small charge relaxation time. Further, the classic Schrödinger equation is obtained when the influence of relaxation time charge is neglected. On the other hand, the complex amplitude of quasi-monochromatic standing waves near the cutoff wavenumber is governed by a similarly type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation makes it possible to estimate the nonlinear effect on the cutoff wavenumber. The nonlinear theory, when used to investigate the stability of charged liquid jet, appears accurately to predict a new unstable regions. The effects of the surface charge and charge relaxation on the stability are identified. The various stability criteria are discussed both analytically and numerically and the stability diagrams are obtained.     

Collapse of solitary waves near the transition from supercritical to subcritical bifurcations

The nonlinear stage of the instability of one-dimensional solitons within a small vicinity of the transition point from supercritical to subcritical bifurcations has been studied both analytically and numerically using the generalized nonlinear Schrödinger equation. It is shown that the pulse amplitude and its width near the collapsing time demonstrate a self-similar behavior with a small asymmetry at the pulse tails due to self-steepening. This theory is applied to solitary interfacial deep-water waves, envelope water waves with a finite depth, and short optical pulses in fibers.     

Counterpropagation of waves with shock fronts in a nonlinear tissue-like medium

A numerical model for describing the counterpropagation of one-dimensional waves in a nonlinear medium with an arbitrary power law absorption and corresponding dispersion is developed. The model is based on general one-dimensional Navier-Stokes equations with absorption in the form of a time-domain convolution operator in the equation of state. The developed algorithm makes it possible to describe wave interactions in the presence of shock fronts in media like biological tissue. Numerical modeling is conducted by the finite difference method on a staggered grid; absorption and sound speed dispersion are taken into account using the causal memory function. The developed model is used for numerical calculations, which demonstrate the absorption and dispersion effects on nonlinear propagation of differently shaped pulses, as well as their reflection from impedance acoustic boundaries.     

A gas kinetic scheme for the Baer–Nunziato two-phase flow model

Numerical methods for the Baer–Nunziato (BN) two-phase flow model have attracted much attention in recent years. In this paper, we present a new gas kinetic scheme for the BN two-phase flow model containing non-conservative terms in the framework of finite volume method. In the view of microscopic aspect, a generalized Bhatnagar–Gross–Krook (BGK) model which matches with the BN model is constructed. Based on the integral solution of the generalized BGK model, we construct the distribution functions at the cell interface. Then numerical fluxes can be obtained by taking moments of the distribution functions, and non-conservative terms are explicitly introduced into the construction of numerical fluxes. In this method, not only the complex iterative process of exact solutions is avoided, but also the non-conservative terms included in the equation can be handled well.     

Numerical Studies of Vanka-Type Smoothers in Computational Solid Mechanics

In this paper multigrid smoothers of Vanka-type are studied in the context of Computational Solid Mechanics (CSM). These smoothers were originally developed to solve saddle-point systems arising in the field of Computational Fluid Dynamics (CFD), particularly for incompressible flow problems. When treating (nearly) incompressible solids, similar equation systems arise so that it is reasonable to adopt the 'Vanka idea' for CSM. While there exist numerous studies about Vanka smoothers in the CFD literature, only few publications describe applications to solid mechanical problems. With this paper we want to contribute to closing this gap. We depict and compare four different Vanka-like smoothers, two of them are oriented towards the stabilised equal-order $Q_1/Q_1$ finite element pair. By means of different test configurations we assess how far the smoothers are able to handle the numerical difficulties that arise for nearly incompressible material and anisotropic meshes. On the one hand, we show that the efficiency of all Vanka-smoothers heavily depends on the proper parameter choice. On the other hand, we demonstrate that only some of them are able to robustly deal with more critical situations. Furthermore, we illustrate how the enclosure of the multigrid scheme by an outer Krylov space method influences the overall solver performance, and we extend all our examinations to the nonlinear finite deformation case.     

Nonlinear waves in media with fifth order dispersion

Nonlinear waves described of the fifth order dispersive nonlinear evolution equation are numerically investigated. The numerical method for boundary value problem for this equation is proposed. Exact solutions to nonlinear evolution equation of the fifth order are given. The numerical method was tested using some exact solutions. The influence of the fifth order dispersion on the propagation of nonlinear waves and formation of the periodic structures is studied.     

Diffusion in a bistable potential: A comparative study of different methods of solution

The problem of diffusion in a bistable potential is studied by considering the associated nonlinear Langevin equation and its equivalent Fokker-Planck equation. Two numerically exact methods of solution, namely, the Monte Carlo solution of the nonlinear Langevin equation and the solution of the Fokker-Planck equation via the finite difference technique, are considered. The latter method has the advantage that it directly gives the evolution of the probability distribution function. Approximate analyses of the fluctuations using the system size expansion, the Gaussian decoupling procedure, and the scaling approach are also carried out. These investigations are performed on a representative problem for two specific cases: (1) evolution from intrinsically unstable states and (2) evolution from extensive regime. The fluctuations obtained using these approximate methods are compared with those obtained via the numerically exact methods. The study brings out the advantages and limitations of each of the methods considered.     

Numerical solutions and analysis of diffusion for new generalized fractional advection-diffusion equations

In this paper we study a class of new Generalized Fractional Advection-Diffusion Equations (GFADEs) with a new Generalized Fractional Derivative (GFD) proposed last year. The new GFD is defined in the Caputo sense using a weight function and a scale function. The GFADE is discussed in a bounded domain, and numerical solutions for two examples consisting of a linear and a nonlinear GFADE are obtained using an implicit finite difference approach. The stability of the numerical scheme is investigated, and the order of convergence is estimated numerically. Numerical results illustrate that the finite difference scheme is simple and effective for solving the GFADEs. We investigate the influence of weight and scale functions on the diffusion of GFADEs. Linear and nonlinear stretching and contracting functions are considered. It is found that an increasing weight function increases the rate of diffusion, and a scale function can stretch or contract the diffusion on the time domain.     

Hamiltonian systems with indefinite kinetic energy

Some interesting features of a class of two-dimensional Hamiltonians with indefinite kinetic energy are considered. It is shown that such Hamiltonians cannot be reduced, in general, to an equivalent dynamical Hamiltonian with positive definite kinetic energy quadratic in velocities. Complex nonlinear evolution equations like the K-dV, the MK-dV and the sine-Gordon equations possess such Hamiltonians. The case of complex K-dV equation has been considered in detail to demonstrate the generic features. The two-dimensional real systems obtained by analytic continuation to complex plane of one-dimensional dynamical systems are also discussed. The evolution equations for nonlinear, amplitude-modulated Langmuir waves as well as circularly polarized electromagnetic waves in plasmas, are considered as illustrative examples.     

Chaotic Behavior of One-Dimensional Nonlinear Schrodinger Equations Involving High Order Saturable Nonlinear Terms

The non-integrable behavior in one-dimensional generalized nonlinear Schr6dinger equations involving high order saturable nonlinearities, which govern the dynamic behavior in a class of physical systems, is investigated numerically. The numerical results illustrate that the high order saturable nonlinearities would lead to chaos, where the irregular homoclinic orbit (HMO) crossings have been observed in phase space. On the other hand, we show that the periodic solutions and solitary waves may exist in such non-integrable continuum Hamiltonian dynamic systems.     

Finite volume treatment of dispersion-relation-preserving and optimized prefactored compact schemes for wave propagation

In developing suitable numerical techniques for computational aero-acoustics, the dispersion-relation-preserving (DRP) scheme by Tam and co-workers and the optimized prefactored compact (OPC) scheme by Ashcroft and Zhang have shown desirable properties of reducing both dissipative and dispersive errors. These schemes, originally based on the finite difference, attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre-determined formal orders of accuracy. In the present study, finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications. Linear and nonlinear wave equations, with and without viscous dissipation, have been adopted as the test problems. Highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases, the performance of these approaches is documented. For the linear wave equation, there is no major difference between the DRP and OPC schemes. For the nonlinear wave equations, the finite volume version of both DRP and OPC schemes offers substantially better solutions in regions of high gradient or discontinuity.     

辐射流体力学的分子动理学自适应加密方法(英文)

通过构造新的平衡分布函数和结合分区自适应网格加密方法,对不带扩散项的平衡辐射流体力学方程,构造二阶的分子动理学BGK-AMR格式.一方面在关心的计算区域中局部加密计算网格,提高计算精度的同时大大节省了计算网格数量和计算时间;另一方面,不同于已有的参数强耦合平衡分布函数,新构造的平衡分布函数中各参数不相互依赖,简化了辐射流体力学分子动理学格式的计算.一维和二维的数值算例显示了格式的性能.     

A High-Order Conservative Numerical Method for Gross-Pitaevskii Equation with Time-Varying Coefficients in Modeling BEC

We propose a high-order conservative method for the nonlinear Schrodinger/Gross-Pitaevskii equation with time-varying coefficients in modeling Bose-Einstein condensation(BEC). This scheme combined with the sixth-order compact finite difference method and the fourth-order average vector field method, finely describes the condensate wave function and physical characteristics in some small potential wells. Numerical experiments are presented to demonstrate that our numerical scheme is efficient by the comparison with the Fourier pseudo-spectral method.Moreover, it preserves several conservation laws well and even exactly under some specific conditions.