A Comparison Study of Three CESE Schemes in MHD Simulation

The space-time conservation element and solution element （CESE） scheme based on the concept of space-time conservation integration scheme is a new second order numerical In order to further overcome excessive numerical damping due to small Courant-Friedrichs-Lewy （CFL） number and to obtain a high quality solution, a Courant number insensitive （CNIS） scheme and a high-order scheme have been proposed by Chang et al. for fluid mechanics problems recently. In this study, to explore the potential capability of applications of the CNIS CESE scheme and the high-order CESE scheme to magnetohydrodynamics （MHD） equations, several benchmark MHD problems are calculated in one and two dimensions： （i） Brio and Wu＇s shock tube, （ii） Dal and Woodward＇s case, （iii） the Orszag-Tang vortex problem, （iv） the Riemann problem. The numerical results just prove that the CNIS scheme is more accurate and can keep the divergence free condition of the magnetic field, even if the CFL number is 〈〈 1. Meanwhile, the tests show that the high order CESE scheme possesses the ability to solve MHD problems but is sensitive to the Courant number.     

Two Types of New Solutions to KdV Equation

It is common knowledge that the soliton solutions u（x, t） defined by the bell-shape form is required to satisfy the following condition lira u（x, t） = u（±∞, t） = 0. However, we think that the above condition can be modified as lim u（x, t） = u（±∞, t）^x→ = c, where c is a constant, which is called as a stationary height of u（x, t） in the present paper.^x→∞ If u（x, t） is a bell-shape solitary solution, then the stationary height of each solitary wave is just c. Under the constraint c = 0, all the solitary waves coming from the N-bell-shape-sollton solutions of the KdV equation are the same-oriented travelling. A new type of N-soliton solution with the bell shape is obtained in the paper, whose stationary height is an arbitrary constant c. Taking c ≥ 0, the resulting solitary wave is bound to be the same-oriented travelling. Otherwise, the resulting solitary wave may travel at the same orientation, and also at the opposite orientation. In addition, another type of singular rational travelling solution to the KdV equation is worked out.     

A Riemann–Hilbert Approach to Complex Sharma–Tasso–Olver Equation on Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value problem of a complex Sharma–Tasso–Olver(c STO) equation on the half line. We show that the solution can be expressed in terms of the solution of a Riemann–Hilbert problem. The relevant jump matrices are explicitly given in terms of the matrix-value spectral functions spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which depending on initial data u_0(x) = u(x, 0) and boundary data g_0(y) = u(0, y), g_1(y) = ux(0, y), g_2(y) = u_(xx)(0, y). These spectral functions are not independent, they satisfy a global relation.     

Theoretical Constraint on Purely Kinetic κ-Essence

Purely kinetic k-essence models in which the Lagrangian contains only a kinetic factor and does not depend explicitly on the field itself are considered, and a theoretical constraint is obtained： Fx -= F0a^-3. Under this theoretical constraint, we discuss a kind of purely κ-essence with form F（X） = -（1 ＋ 2X^n）^1/2n, which can be considered as the generalized tachyon field, and find that this kind of κ-essence is not likely a candidate of dark energy to describe the present accelerated expansion of the Universe. This is contrary to a previous suggestion that κ-essence with such a form may be used to describe phantom cosmologies.     

Separability of Solutions to a Schr?dinger Equation

We discuss separability of solutions to a Schr?dinger equation that describes a composite quantum system and give some kinds of Hamiltonians H(t) such that the solution to Schr?dinger equation induced by H(t) is separable at any time provided that it is separable at t = 0. For example, we prove that if the Hamiltonian H is time-independent and equals to the product PA■PB of two projections on the subsystems KAand KB, respectively, then the state |ψ(t) of the composite system starting from a separable initial |ψ(0) = |ψA■|ψB is separable for all t ∈ [0, T] if and only if either |ψA is an eigenstate of PA, or |ψB is an eigenstate of PB.     

Analysis of variable coefficient advection-diffusion problems via complex variable reproducing kernel particle method

The complex variable reproducing kernel particle method （CVRKPM） of solving two-dimensional variable coefficient advection-diffusion problems is presented in this paper. The advantage of the CVRKPM is that the shape function of a two-dimensional problem is formed with a one-dimensional basis function. The Galerkin weak form is employed to obtain the discretized system equation, and the penalty method is used to apply the essential boundary conditions. Then the corresponding formulae of the CVRKPM for two-dimensional variable coefficient advection-diffusion problems are obtained. Two numerical examples are given to show that the method in this paper has greater accuracy and computational efficiency than the conventional meshless method such as reproducing the kernel particle method （RKPM） and the element- free Galerkin （EFG） method.     

Extra Dimensions and Vacuum Dark Energy Models

The role of vacuum energy or cosmological constant in cosmology is discussed in a kind of nontrivial higherdimensional model. Under the framework of Einstein‘s gravity, we obtain the corresponding equations of motion and find that the cosmological constant and vacuum energy in the full regime does not drive its acceleration, but decelerates the expansion of the universe. The dimension of space is required to be n = 3 if we regard vacuum energy or cosmological constant as the candidate to drive the accelerated expansion of the universe.     

A three-dimensional Eulerian method for the numerical simulation of high-velocity impact problems

In the present paper, a three-dimensional （3D） Eulerian technique for the 3D numerical simulation of high-velocity impact problems is proposed. In the Eulerian framework, a complete 3D conservation element and solution element scheme for conservative hyperbolic governing equations with source terms is given. A modified ghost fluid method is proposed for the treatment of the boundary conditions. Numerical simulations of the Taylor bar problem and the ricochet phenomenon of a sphere impacting a plate target at an angle of 60~ are carried out. The numerical results are in good agreement with the corresponding experimental observations. It is proved that our computational technique is feasible for analyzing 3D high-velocity impact problems.     

Invariant Sets and Exact Solutions to Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.     

N-soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method

In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.     

Multi-objective optimization of inverse planning for accurate radiotherapy

The multi-objective optimization of inverse planning based on the Pareto solution set, according to the multi-objective character of inverse planning in accurate radiotherapy, was studied in this paper. Firstly, the clinical requirements of a treatment plan were transformed into a multi-objective optimization problem with multiple constraints. Then, the fast and elitist multi-objective Non-dominated Sorting Genetic Algorithm (NSGA-Ⅱ) was introduced to optimize the problem. A clinical example was tested using this method. The results show that an obtained set of non-dominated solutions were uniformly distributed and the corresponding dose distribution of each solution not only approached the expected dose distribution, but also met the dose-volume constraints. It was indicated that the clinical requirements were better satisfied using the method and the planner could select the optimal treatment plan from the non-dominated solution set.     

Initial-Boundary Value Problem for Two-Component Gerdjikov–Ivanov Equation with 3 × 3 Lax Pair on Half-Line

The Fokas unified method is used to analyze the initial-boundary value problem of two-component Gerdjikov–Ivanonv equation on the half-line. It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 × 3 Riemann–Hilbert problem. The Dirichlet to Neumann map is obtained through the global relation.     

Aerodynamic design optimization by using a continuous adjoint method

This paper presents the fundamentals of a continuous adjoint method and the applications of this method to the aerodynamic design optimization of both external and internal flows.General formulation of the continuous adjoint equations and the corresponding boundary conditions are derived.With the adjoint method,the complete gradient information needed in the design optimization can be obtained by solving the governing flow equations and the corresponding adjoint equations only once for each cost function,regardless of the number of design parameters.An inverse design of airfoil is firstly performed to study the accuracy of the adjoint gradient and the effectiveness of the adjoint method as an inverse design method.Then the method is used to perform a series of single and multiple point design optimization problems involving the drag reduction of airfoil,wing,and wing-body configuration,and the aerodynamic performance improvement of turbine and compressor blade rows.The results demonstrate that the continuous adjoint method can efficiently and significantly improve the aerodynamic performance of the design in a shape optimization problem.     

Suppression of spiral waves using intermittent local electric shock

In this paper, an intermittent local electric shock scheme is proposed to suppress stable spiral waves in the Barkley model by a weak electric shock (about 0.4 to 0.7) imposed on a random selected n×n grids (n=1-5, compared with the original 256×256 lattice) and monitored synchronically the evolutions of the activator on the grids as the sampled signal of the activator steps out a given threshold (i.e., the electric shock works on the n×n grids if the activator u\leq0.4 or u \geq 0.8). The numerical simulations show that a breakup of spiral is observed in the media state evolution to finally obtain homogeneous states if the electric shock with appropriate intensity is imposed.     

Investigation of Resistivity of Saturated Porous Media with Lattice Boltzmann Method

The lattice Boltzmann method is employed to study the electrical transport properties of saturated porous media.Electrical current flow through the porous media is simulated and the relationship between resistivity index and water saturation is derived. It is found that this kind of relation is not a straight line as described by the Archie equation with the parameter n being a constant in a log-log scale. A new equation is thus developed to formulate this relation with n being a function of porosity and water saturation. The comparisons between the results by lattice Boltzmann and by the laboratory experiments on rock samples demonstrate that this numerical method can provide an alternative way for the expensive laboratory experiments to investigate the electrical transport properties of saturated porous media and can be used to explore micro mechanisms more conveniently.     

Laser-Manipulating Macroscopic Quantum States of a Bose-Einstein Condensate Held in a Kronig-Penny Potential

We investigate the boundary vaJue problem （BVP） of a quasi-one-dimensional Gross-Pitaevskii equation with the Kronig-Penney potential （KPP） of period d, which governs a repulsive Bose-Einstein condensate. Under the zero and periodic boundary conditions, we show how to determine n exact stationary eigenstates {Rn} corresponding to different chemical potentials {μn} from the known solutions of the system. The n-th eigenstate P~ is the Jacobian elliptic function with period 2din for n = 1,2,…, and with zero points containing the potential barrier positions. So Rn is differentiable at any spatial point and R2 describes n complete wave-packets in each period of the KPP. It is revealed that one can use a laser pulse modeled by a 5 potential at site xi to manipulate the transitions from the states of {Rn} with zero Point x≠xi to the states of {Rn＇} with zero Point x= Xi. The results suggest an experimental scheme for applying BEC to test the BVP and to observe the macroscopic quantum transitions.     

Exactly solving some typical Riemann–Liouville fractional models by a general method of separation of variables

Finding exact solutions for Riemann–Liouville(RL) fractional equations is very difficult. We propose a general method of separation of variables to study the problem. We obtain several general results and, as applications, we give nontrivial exact solutions for some typical RL fractional equations such as the fractional Kadomtsev–Petviashvili equation and the fractional Langmuir chain equation. In particular, we obtain non-power functions solutions for a kind of RL time-fractional reaction–diffusion equation. In addition, we find that the separation of variables method is more suited to deal with high-dimensional nonlinear RL fractional equations because we have more freedom to choose undetermined functions.     

Transient Dynamics of Light Propagation in A-Atom EIT Medium

We investigate the transient phenomenon or property of the propagation of an optical probe field in a medium consisting of many A-type three-level atoms coupled to this probe field and a classical driven field. We observe a hidden symmetry and obtain an exact solution for this light propagation problem by means of the spectral generating method. This solution enlightens us to propose a practical protocol implementing the quantum memory robust for quantum decoherence in a crystal. As a transient dynamic process this solution also manifests an exotic result that a wave-packet of light will split into three packets propagating at different group velocities. It is argued that “super-luminal group velocity” and “sub-luminal group velocity” can be observed simultaneously in the same system. This interesting phenomenon is expected to be demonstrated experimentally.     

Global Solutions of Einstein—Dirac Equation on the Conformal Space

The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic.Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group,which is isomorphic to SL(2,C).Hence the results and their interpretation coming from the two theories would be different.In this short note we study only the Lorentz spinor manifold and,especially,the solutions of Einstein-Dirac equations on the conformal space,which is closely related to the AdS/CFT correspondence.     

New approach to the normal mode method in underwater acoustics

A new approach to the numerical solution of normal model problems in underwater acoustics is presented,in which the corresponding normal mode problem is transformed to the problem of solving a dynamic system.Three applications are considered:(1)the broad band normal mode problem;(2) the range-dependent problem with perturbation proportional to the range parameter;and (3) the evolution of the normal mode with environmental parameters.A numerical simulation for a broad band problem is performed,and the calculated eigenvalues have good agreement with those obtained by the standard normal mode code KRAKAN.