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In this paper, we propose and analyze two conservative fourth-order compact finite difference schemes for the (1+1) dimensional nonlinear Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise forms of the proposed compact schemes into equivalent vector forms and analyze their conservative and convergence properties. We prove that the proposed schemes preserve the total mass and energy in the discrete level and the convergence rate of the schemes, without any restrictions on the grid ratio, are at the order of O ( h 4 + τ 2 ) in l ∞ -norm, where h and τ are spatial and temporal steps, respectively. The error analysis techniques include the energy method and the techniques of either the cut-off of the nonlinearity or the mathematical induction to bound the numerical approximate solutions. The numerical experiments are carried out to confirm our theoretical analysis. The research method in this paper can be easily extended to higher order compact schemes or other types of wave equations.

In this study, we design and analyse an efficient and accurate Fourier pseudo-spectral method for solving the nonlinear Schrodinger equation with wave operator (NLSW). In this method, a modified le...

In this paper, we study two compact finite difference schemes for the Schrodinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H1 error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ2 + h4) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrodinger–Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O ( τ 2 + h 4 ) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis.

The objective of this paper is to propose some high-order compact schemes for two-dimensional Ginzburg-Landau equation. The space is approximated by high-order compact methods to improve the computational efficiency. Based on Crank-Nicolson method in time, several temporal approximations are used starting from different viewpoints. The numerical characters of the new schemes, such as the existence and uniqueness, stability, convergence are investigated. Some numerical illustrations are reported to confirm the advantages of the new schemes by comparing with other existing works. In the numerical experiments, the role of some parameters in the model is considered and tested.

Two numerical methods are presented for the approximation of the Zakharov-Rubenchik equations (ZRE). The first one is the finite difference integrator Fourier pseudospectral method (FFP), which is implicit and of the optimal convergent rate at the order of O(N−r + τ2) in the discrete L2 norm without any restrictions on the grid ratio. The second one is to use the Fourier pseudospectral approach for spatial discretization and exponential wave integrator for temporal integration. Fast Fourier transform is applied to the discrete nonlinear system to speed up the numerical computation. Numerical examples are given to show the efficiency and accuracy of the new methods.

In this paper, we propose and analyze a conservative fourth-order compact finite difference scheme for the Klein–Gordon–Dirac equation with periodic boundary conditions. Based on matrix knowledge, we convert the point-wise form of the proposed compact scheme into equivalent vector form and analyze its conservative and convergence properties. We prove that the new scheme conserves the total mass and energy in the discrete level and the convergence rate of the scheme, without any restrictions on the grid ratio, is at the order of $$O(h^4 +\tau ^2)$$ in $$l^\infty $$ -norm, where h and $$\tau $$ are spatial and temporal steps, respectively. The techniques for error analysis include the energy method and the mathematical induction. The numerical experiments are carried out to confirm our theoretical analysis.

The rapid development of refining and chemical enterprises has brought great support to the national economy, but the outstanding security problems of them have also attracted great attention. How to scientifically and accurately evaluate the safety level of refining and chemical enterprises has become a hot topic. The advent of the era of big data has also brought about changes in the study of Safety Level Assessment (SLA) technique. The growth of information volume, the development of big data analysis and computing technology will help to overcome the shortcomings of traditional SLA technologies from a statistical point of view, thus forming a more accurate one. In the above context, this paper focus on the impact of the data on the development of SLA technology and propose the connotation and construction ideas of the data-driven SLA system. The network crawler technology, entropy weight method, Page-Rank algorithm are used to build a scientific and perfect refining and chemical enterprise SLA system, in terms of data collection, index setting, index weight calculation. The limitations of this research and the future direction are also analyzed in this paper. The research results are applied in four refining and chemical enterprises. The application results show that the research results can effectively complete the horizontal comparison between enterprises, reveal the problems and loopholes in management, and it is worth further development and promotion.

In view of the problems such as the large workload of video monitoring on drilling site and the lack of effective utilization of massive video data, firstly, the risk-based video monitoring layout optimization method was put forward by using particle swarm optimization algorithm after comprehensive consideration of the monitoring area coverage, key monitoring in high-risk areas and cost and other factors. Based on the identification and screening of high-risk behaviors with high accident consequences, the overflow monitoring scene was selected to design a deep neural network based on Faster RCNN and OpenPose frame to identify the arrival of personnel and squat sampling actions. Video intelligent analysis technology was developed, and video intelligent analysis and alarm system was developed to carry out real-time behavior detection of onsite overflow monitoring process. The results show that the human sampling motion recognition accuracy is 85.6%, and the system detection rate is 130 ms/frame, achieving good practical results.

This paper is concerned with the time-step condition of linearized implicit finite difference method for solving the Gross-Pitaevskii equation with an angular momentum rotation term. Unlike the existing studies in the literature, where the cut-off function technique was used to establish the error estimates under some conditions of the time-step size, this paper introduces an induction argument and a ‘lifting’ technique as well as some useful inequalities to build the optimal maximum error estimate without any constraints on the time-step size. The analysis method can be directly extended to the general nonlinear Schrodinger-type equations in twoand three-dimensions and other linear implicit finite difference schemes. As a by-product, this paper defines a new type of energy functional of the grid functions by using a recursive relation to prove that the proposed scheme preserves well the total mass and energy in the discrete sense. Several numerical results are reported to verify the error estimates and conservation laws.

It is difficult for traditional risk decision-making methods to simulate the strategic interaction between the pipeline company and the third party, which restricts the effective allocation of pipeline defensive resources. In view of this, the game theory (GT) combined with traditional risk assessment (TRA) method is used to analyze the interaction process. Considering different attack targets, attack strategies, and differences in revenue calculations of different third party types, the game model is extended into a Bayesian-Stackelberg model under the assumption of complete rationality and common knowledge. The proposed method is verified on an oil pipeline. The return on investment can be 15.14 and shows that this approach can give advice to the optimization of defense resource allocation to achieving the purpose of reducing operation and maintenance costs by comparing the expected losses of different defense strategies by assignment calculations.

The coupled Gross–Pitaevskii system studied in this paper is an important mathematical model describing spin-1 Bose-Einstein condensate. We propose a linearized and decoupled compact finite difference scheme for the coupled Gross–Pitaevskii system, which means that only three tri-diagonal systems of linear algebraic equations at each time step need to be solved by using Thomas algorithm. New types of mass functional, magnetization functional and energy functional are defined by using a recursive relation to prove that the new scheme preserves the total mass, energy and magnetization in the discrete sense. Besides the standard energy method, we introduce an induction argument as well as a lifting technique to establish the optimal error estimate of the numerical solution without imposing any constraints on the grid ratios. The convergence order of the new scheme is of O ( h 4 + τ 2 ) in the L2 norm and H1 norm, respectively, with time step τ and mesh size h. Our analysis method can be used to high dimensional cases and other linearized finite difference schemes for the two- or three-dimensional nonlinear Schrodinger/Gross–Pitaevskii equations. Finally, numerical results are reported to test the theoretical results.

This paper is concerned with the time-step condition (or grid ratio condition) of finite difference method for solving the Gross–Pitaevskii equation with an angular momentum rotation term. An optimal error estimate of the finite difference method in H 1 norm is established without any constraints of the grid ratios. Besides the standard energy method, a ‘cut-off’ function technique and a ‘lifting’ technique is introduced in analyzing the proposed scheme. The analysis method used in this paper can be applied to many other finite difference schemes for solving the nonlinear Schrodinger-type equations for which previous works often require certain restriction on the grid ratios. Numerical results are reported to verify the error estimates and conservation laws.

It has been pointed out in literature that the symplectic scheme of a nonlinear Hamiltonian system can not preserve the total energy in the discrete sense Ge and Marsden (1988) [10] . Moreover, due to the difficulty in obtaining a priori estimate of the numerical solution, it is very hard to establish the optimal error bound of the symplectic scheme without any restrictions on the grid ratios. In this paper, we develop and analyze a compact scheme for solving nonlinear Schrodinger equation. We introduce a cut-off technique for proving optimal L ∞ error estimate for the compact scheme. We show that the convergence of the compact scheme is of second order in time and of fourth order in space. Meanwhile, we define a new type of energy functional by using a recursion relationship, and then prove that the compact scheme is mass and energy-conserved, symplectic-conserved, unconditionally stable and can be computed efficiently. Numerical experiments confirm well the theoretical analysis results.

The management of hot work is the key element of petroleum refinery safety management. A large amount of hot-work data has been accumulated, which is underexploited. Driving new insights using big data analytics is the trend. However, there have been few scientific studies on solving a specific problem using big data method in the field of safety management. Hence, the unstructured-data analysis of the hot-work permit-to-work text was investigated. The professional corpus in the hot-work field was constructed using word segmentation, stop list elimination, standardized process, and manual proofreading. All hot-work content collocates were manually grouped into semantic domains in the light of experts’ experience. The data-driven reduction models of hot work number were proposed aiming at possible invalid hot work, long-time hot work, repetitive hot work and possible equipment defect. Based on the proposed reduction models, we mined the patterns of hot work and the unnecessary or high-risk hot work could be identified automatically. The result of the reduction model in training set indicated that the reduction models are reasonable. The reduction model 1, reduction model 2, reduction model 3, reduction model 4 could reduce the proportion of hot work number 5%, 3%, 4%, and 2%, respectively. Thus, the targeted measures could be put forward to optimize the safety management of hot work.

In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrodinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an H1 technique are introduced to establish the optimal point-wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order of the numerical solution is proved to be of O ( h 4 + τ 2 ) with mesh size h and time step τ. Numerical results are reported to verify the theoretical analysis, and collision of two solitary waves are also simulated.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2016

This paper aims to construct and analyze two new Fourier pseudo-spectral (FPS) methods for the general nonlinear Schrodinger (NLS) equation. The two FPS methods have two merits: unconditional convergence and complete explicitness in the practical computation. Further more, by introducing a modified mass functional and a modified energy functional, the two FPS methods are proved to preserve the total mass and energy in the discrete sense. Besides the standard energy method, the key techniques used in our numerical analysis are a mathematical induction argument and a lifting technique. Without any restriction on the grid ratio and initial value, we establish the optimal error estimate of the two FPS methods for solving the general NLS equation, while previous work just is valid for the cubic NLS equation and requires small initial value for the focusing case. These two FPS methods are proved to be spectrally accurate in space and second-order accurate in time, respectively. The analysis framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the NLS-type equations. We investigate the effect of the nonlinear term on the progression simulation of the plane wave, the conservation of the invariants and the effect of initial data on the blow-up solution via different parameters. Numerical results are reported to show the accuracy and efficiency of the proposed methods.

This paper concerns the null controllability of a system ofmlinear degenerate parabolic equations with coupling terms of first and zero order, and only one control force localized in some arbitrary nonempty open subsetωof Ω. The key ingredient for proving the null controllability is to obtain the observability inequality for the corresponding adjoint system. Due to the degeneracy, we transfer to study an approximate nondegenerate adjoint system. In order to deal with the coupling first order terms, we first prove a new Carleman estimate for a degenerate parabolic equation in Sobolev spaces of negative order. Based on this Carleman estimate, we obtain a uniform Carleman estimate and then an observation inequality for this approximate adjoint system.

This paper is concerned with the unconditional and optimal $$L^{\infty }$$ -error estimates of two fourth-order (in space) compact conservative finite difference time domain schemes for solving the nonlinear Schrodinger equation in two or three space dimensions. The fact of high space dimension and the approximation via compact finite difference discretization bring difficulties in the convergence analysis. The two proposed schemes preserve the total mass and energy in the discrete sense. To establish the optimal convergence results without any constraint on the time step, besides the standard energy method, the cut-off function technique as well as a ‘lifting’ technique are introduced. On the contrast, previous works in the literature often require certain restriction on the time step. The convergence rate of the proposed schemes are proved to be of $$O(h^4+\tau ^2)$$ with time step $$\tau $$ and mesh size h in the discrete $$L^{\infty }$$ -norm. The analysis method can be directly extended to other finite difference schemes for solving the nonlinear Schrodinger-type equations. Numerical results are reported to support our theoretical analysis, and investigate the effect of the nonlinear term and initial data on the blow-up solution.

The focus of this paper is on the optimal error bounds of two finite difference schemes for solving the d-dimensional (d = 2, 3) nonlinear Klein-Gordon-Schrodinger (KGS) equations. The proposed finite difference schemes not only conserve the mass and energy in the discrete level but also are efficient in practical computation because only two linear systems need to be solved at each time step. Besides the standard energy method, an induction argument as well as a ‘lifting’ technique are introduced to establish rigorously the optimal H 2-error estimates without any restrictions on the grid ratios, while the previous works either are not rigorous enough or often require certain restriction on the grid ratios. The convergence rates of the proposed schemes are proved to be at O(h 2 + τ 2) with mesh-size h and time step τ in the discrete H 2-norm. The analysis method can be directly extended to other linear finite difference schemes for solving the KGS equations in high dimensions. Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.

This study is devoted to analysis of semi-implicit compact finite difference (SICFD) methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter e ∈ ( 0 , 1 ] . Uniform l ∞ -norm error bounds of the proposed SICFD schemes are built to give immediate insight on point-wise error occurring as time increases, and the explicit dependence of the mesh size and time step on the parameter e is also figured out. In the small e regime, highly oscillations arise in time with O ( e 2 ) -wavelength. This highly oscillatory nature in time as well as the difficulty raised by the compact FD discretization make establishing the l ∞ -norm error bounds uniformly in e of the SICFD methods for NLSW to be a very interesting and challenging issue. The uniform l ∞ -norm error bounds in e are proved to be of O ( h 4 + τ ) and O ( h 4 + τ 2 / 3 ) with time step τ and mesh size h for well-prepared and ill-prepared initial data. Finally, numerical results are reported to verify the error estimates and show the sharpness of the convergence rates in the respectively parameter regimes.

The following coupled Schrodinger system with a small perturbation $$\begin{array}{*{20}c} {u_{xx} + u - u^3 + \beta uv^2 + \varepsilon f(\varepsilon ,u,u_x ,v,v_x ) = 0 in \mathbb{R},} \\ {v_{xx} - v + v^3 + \beta u^2 v + \varepsilon g(\varepsilon ,u,u_x ,v,v_x ) = 0 in \mathbb{R}} \\ \end{array}$$ is considered, where β and ∈ are small parameters. The whole system has a periodic solution with the aid of a Fourier series expansion technique, and its dominant system has a heteroclinic solution. Then adjusting some appropriate constants and applying the fixed point theorem and the perturbation method yield that this heteroclinic solution deforms to a heteroclinic solution exponentially approaching the obtained periodic solution (called the generalized heteroclinic solution thereafter).

In this paper, we analyze a compact finite difference scheme for computing a coupled nonlinear Schrodinger equation. The proposed scheme not only conserves the total mass and energy in the discrete level but also is decoupled and linearized in practical computa- tion. Due to the difficulty caused by compact difference on the nonlinear term, it is very hard to obtain the optimal error estimate without any restriction on the grid ratio. In order to overcome the difficulty, we transform the compact difference scheme into a special and equivalent vector form, then use the energy method and some important lemmas to obtain the optimal convergent rate, without any restriction on the grid ratio, at the order of O(h 4 +� 2 ) in the discrete L 1 -norm with time stepand mesh size h. Finally, numerical results are reported to test our theoretical results of the proposed scheme. Mathematics subject classification: 65M06, 65M12.

In this paper, we propose a compact finite difference scheme for computing the Klein–Gordon–Schrodinger equation (KGSE) with homogeneous Dirichlet boundary conditions. The proposed scheme not only conserves the total mass and energy in the discrete level but also is linearized in practical computation. Except for the standard energy method, a new technique is introduced to obtain the optimal convergent rate, without any restriction on the grid ratios, at the order of O ( h 4 + τ 2 ) in the l ∞ -norm with time step τ and mesh size h . Finally, numerical results are reported to test the theoretical results.

Due to the difficulty in obtaining the a priori estimate, it is very hard to establish the optimal point-wise error bound of a finite difference scheme for solving a nonlinear partial differential equation in high dimensions (2D or 3D). We here propose and analyze finite difference methods for solving the coupled Gross-Pitaevskii equations in two dimensions, which models the two-component Bose-Einstein condensates with an internal atomic Josephson junction. The methods which we considered include two conservative type schemes and two non-conservative type schemes. Discrete conservation laws and solvability of the schemes are analyzed. For the four proposed finite difference methods, we establish the optimal convergence rates for the error at the order of O(h 2 +τ 2) in the l ∞-norm (i.e., the point-wise error estimates) with the time step τ and the mesh size h. Besides the standard techniques of the energy method, the key techniques in the analysis is to use the cut-off function technique, transformation between the time and space direction and the method of order reduction. All the methods and results here are also valid and can be easily extended to the three-dimensional case. Finally, numerical results are reported to confirm our theoretical error estimates for the numerical methods.

In this paper, various split step spectral (SSSP) schemes are proposed for N-coupled nonlinear Schrodinger (N-CNLS) equations, especially for the systems with N>=3 for which numerical studies are few. These schemes are spectrally accurate in space, and sth (s=1,2,4,6,8) order in time. They are proved to be conservative and to admit the exact plane wave solution. Extensive numerical experiments are carried out for the 3- and 4-CNLS systems to confirm the theoretical analysis. Half of the schemes are shown to possess wonderful ability of capturing high-frequency waves. However, the eighth-order schemes which seems to be optimal fail in this test. Accuracy and efficiency of the schemes are compared with each other, and the high-order schemes exhibit better. Since the eighth-order schemes are not better than the sixth-order ones, it is believed that constructing too much high-order schemes are unnecessary. Moreover, interactions of two-soliton solutions are also well simulated by the fourth-order and the sixth-order schemes, so the two methods are sufficient for use.

A compact and semi-explicit finite difference scheme is proposed and analyzed for the Klein-Gordon-Zakharov (KGZ) equation. The new scheme is decoupled and linearized in practical computation, i.e., at each time step only two tri-diagonal systems of linear algebraic equations need to be solved by Thomas algorithm. So the new scheme is more efficient and more accurate than the classical finite difference schemes. Unique solvability of the difference solution is proved by using the energy method. Besides the standard energy method, in order to overcome the difficulty in obtaining the a priori estimate, an induction argument is introduced to prove that the new scheme is convergent for u ( x , t ) in the discrete H 1 -norm, and respectively for m ( x , t ) in the discrete L 2 -norm, at the order of O ( ? 2 + h 4 ) with time step ? and mesh size h. Numerical results are reported to verify the theoretical analysis.

The coupled Gross---Pitaevskii (CGP) equation studied in this paper is an important mathematical model describing two-component Bose---Einstein condensate with an internal atomic Josephson junction. We here analyze a compact finite difference scheme which conserves the total mass and energy in the discrete level for the CGP equation. In general, due to the difficulty caused by compact difference on nonlinear terms, optimal point-wise error estimates without any restrictions on the grid ratios of compact difference schemes for nonlinear partial differential equations are very hard to be established. To overcome the difficulty caused by the compact difference operator, we introduce a new norm and an interesting transformation by which the difference scheme is transformed into a special equivalent vector form, we then use the energy method and some important lemmas on the equivalent system to obtain the optimal convergent rate, without any restrictions on the grid ratio, at the order of $$O(h^{4}+\tau ^2)$$ O ( h 4 + ? 2 ) in the maximum norm with time step $$\tau $$ ? and mesh size $$h$$ h . Finally, numerical results are reported to test the theoretical results and simulate the dynamics of the CGP equation.

In this paper, a fourth-order compact and energy conservative difference scheme is proposed for solving the two-dimensional nonlinear Schrodinger equation with periodic boundary condition and initial condition, and the optimal convergent rate, without any restriction on the grid ratio, at the order of O ( h 4 + τ 2 ) in the discrete L 2 -norm with time step τ and mesh size h is obtained. Besides the standard techniques of the energy method, a new technique and some important lemmas are proposed to prove the high order convergence. In order to avoid the outer iteration in implementation, a linearized compact and energy conservative difference scheme is derived. Numerical examples are given to support the theoretical analysis.