Solar-powered aircraft have attracted great attention owing to their potential for longendurance flight and wide application prospects.Due to the particularity of energy system,flight strategy optimization is a signif...Solar-powered aircraft have attracted great attention owing to their potential for longendurance flight and wide application prospects.Due to the particularity of energy system,flight strategy optimization is a significant way to enhance the flight performance for solar-powered aircraft.In this study,a flight strategy optimization model for high-altitude long-endurance solar-powered aircraft was proposed.This model consists of three-dimensional kinematic model,aerodynamic model,energy collection model,energy store model and energy loss model.To solve the nonlinear optimal control problem with process constraints and terminal constraints,Gauss pseudo-spectral method was employed to discretize the state equations and constraint equations.Then a typical mission flying from given initial point to given final point within a time interval was considered.Results indicate that proper changes of the attitude angle contribute to increasing the energy gained by photovoltaic cells.Utilization of gravitational potential energy can partly take the role of battery pack.Integrating these two measures,the optimized flight strategy can improve the final state of charge compared with current constant-altitude constant-velocity strategy.The optimized strategy brings more profits on condition of lower sunlight intensity and shorter daytime.展开更多
To improve the modeling accuracy of radiative transfer,the scattering properties of aerosol particles with irregular shapes and inhomogeneous compositions should be simulated accurately.To this end,a light-scattering ...To improve the modeling accuracy of radiative transfer,the scattering properties of aerosol particles with irregular shapes and inhomogeneous compositions should be simulated accurately.To this end,a light-scattering model for nonspherical particles is established based on the pseudo-spectral time domain(PSTD)technique.In this model,the perfectly matched layer with auxiliary differential equation(ADE-PML),an excellent absorption boundary condition(ABC)in the finite difference time domain generalized for the PSTD,and the weighted total field/scattered field(TF/SF)technique is employed to introduce the incident light into 3 D computational domain.To improve computational efficiency,the model is further parallelized using the Open MP technique.The modeling accuracy of the PSTD scheme is validated against Lorenz–Mie,Aden–Kerker,T-matrix theory and DDA for spheres,inhomogeneous particles and nonspherical particles,and the influence of the spatial resolution and thickness of ADE-PML on the modeling accuracy is discussed as well.Finally,the parallel computational efficiency of the model is also analyzed.The results show that an excellent agreement is achieved between the results of PSTD and well-tested scattering models,where the simulation errors of extinction efficiencies are generally smaller than 1%,indicating the high accuracy of our model.Despite its low spatial resolution,reliable modeling precision can still be achieved by using the PSTD technique,especially for large particles.To suppress the electromagnetic wave reflected by the absorption layers,a six-layer ADE-PML should be set in the computational domain at least.展开更多
In this article, numerical modeling of borehole radar for well logging in time domain is developed using pseudo-spectral time domain algorithm in axisymmetric cylindrical coordinate for proximate true formation model....In this article, numerical modeling of borehole radar for well logging in time domain is developed using pseudo-spectral time domain algorithm in axisymmetric cylindrical coordinate for proximate true formation model. The conductivity and relative permittivity logging curves are obtained from the data of borehole radar for well logging. Since the relative permittivity logging curve is not affected by salinity of formation water, borehole radar for well logging has obvious advantages as compared with conventional electrical logging. The borehole radar for well logging is a one-transmitter and two-receiver logging tool. The conductivity and relative permittivity logging curves are obtained successfully by measuring the amplitude radio and the time difference of pulse waveform from two receivers. The calculated conductivity and relative permittivity logging curves are close to the true value of surrounding formation, which tests the usability and reliability of borehole radar for well logging. The numerical modeling of borehole radar for well logging laid the important foundation for researching its logging tool.展开更多
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. ...Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.展开更多
With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important me...With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important methods of wave-field simulation. Results of previous studies show that both methods have distinct advantages and disadvantages: Finite difference method has high precision but its dispersion is serious; pseudospectral method considers both computational efficiency and precision but has less precision than finite-difference. The authors consider the complex structural characteristics of the metal ore,furthermore add random media in order to simulate the complex effects produced by metal ore for wave field. First,the study introduced the theories of random media and two forward modelling methods. Second,it compared the simulation results of two methods on fault model. Then the authors established a complex metal ore model,added random media and compared computational efficiency and precision. As a result,it is found that finite difference method is better than pseudo-spectral method in precision and boundary treatment,but the computational efficiency of pseudospectral method is slightly higher than the finite difference method.展开更多
In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependen...In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L<sub>2</sub> and L<sub>∞</sub> error norms are computed to study the accuracy and the simplicity of the presented method.展开更多
In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caput...In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.展开更多
This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogo...This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.展开更多
This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and ...This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and transition of 43×43 computational grids,a coordinate transformation is put up from physical panel to computational panel. Several zero turbulent models are computed comparatively. The results are credible when comparing with the previous methods.展开更多
In this paper, the pseudo-spectral approximations for a class of the Kdv-Burgers type equation is presented. Convergence and stability of the approximation have been proved by Sobolev's inequalities and the bounde...In this paper, the pseudo-spectral approximations for a class of the Kdv-Burgers type equation is presented. Convergence and stability of the approximation have been proved by Sobolev's inequalities and the bounded extensive method of the nonlinear function. Finally, the numerical examples are proposed.展开更多
In the paper,we consider the coupled nonlinear Schrödinger equation with high degree polynomials in the energy functional that cannot be handled by using the newly proposed quadratic auxiliary variable method.The...In the paper,we consider the coupled nonlinear Schrödinger equation with high degree polynomials in the energy functional that cannot be handled by using the newly proposed quadratic auxiliary variable method.Therefore,we develop the multiple quadratic auxiliary variable approach to deal with coupled systems and construct high-accuracy structure-preserving schemes for the equation.To fix the idea,we first apply the multiple quadratic auxiliary variable approach to the equation and obtain an equivalent system that possesses the original energy and mass.Then,a family of high-accuracy structure-preserving schemes that can conserve the mass and energy is derived by applying the Gauss collocation method and sine pseudo-spectral method to approximate the system in time and space.The given schemes have high-accuracy in time and can both inherit the mass and Hamiltonian energy of the system.Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.展开更多
Abstract Recently,the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention.For the nonlinear Schrödinger equation(NLS)with wave operator(NLSW)and weak nonl...Abstract Recently,the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention.For the nonlinear Schrödinger equation(NLS)with wave operator(NLSW)and weak nonlinearity controlled by a small valueε∈(0,1],an exponential wave integrator Fourier pseudo-spectral(EWIFP)discretization has been developed(Guo et al.,2021)and proved to be uniformly accurate aboutεup to the time atΟ(1/ε^(2))However,the EWIFP method is not time symmetric and can not preserve the discrete energy.As we know,the time symmetry and energy-preservation are the important structural features of the true solution and we hope that this structure can be inherited along the numerical solution.In this work,we propose a time symmetric and energy-preserving exponential wave integrator Fourier pseudo-spectral(SEPEWIFP)method for the NLSW with periodic boundary conditions.Through rigorous error analysis,we establish uniform error bounds of the numerical solution atΟ(h^(mo)+ε^(2-βτ2))up to the time atΟ(1/ε^(β))forβ∈[0,2]where h andτare the mesh size and time step,respectively,and m0 depends on the regularity conditions.The tools for error analysis mainly include cut-off technique and the standard energy method.We also extend the results on error bounds,energy-preservation and time symmetry to the oscillatory NLSW with wavelength atΟ(1/ε^(2))in time which is equivalent to the NLSW with weak nonlinearity.Numerical experiments confirm that the theoretical results in this paper are correct.Our method is novel because that to the best of our knowledge there has not been any energy-preserving exponential wave integrator method for the NLSW.展开更多
A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-SchrSdinger equations. The scheme is of spectral accura...A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-SchrSdinger equations. The scheme is of spectral accuracy in space and of second order in time. The scheme preserves the discrete multisymplectic conservation law and the charge conservation law. Moreover, the residuals of some other conservation laws are derived for the geometric numerical integrator. Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme, and demonstrate the correctness of the theoretical analysis.展开更多
Proposes an explicit fully discrete three-level pseudo-spectral scheme with unconditional stability for the Cahn-Hilliard equation. Equations for pseudo-spectral scheme; Analysis of linear stability of critical points.
This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the n...This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.展开更多
In this paper,we propose a novel noncausal control framework to address the energy maximization problem of wave energy converters(WECs)subject to constraints.The energy maximization problem of WECs is a constrained op...In this paper,we propose a novel noncausal control framework to address the energy maximization problem of wave energy converters(WECs)subject to constraints.The energy maximization problem of WECs is a constrained optimal control problem.The proposed control framework converts this problem into a reference trajectory tracking problem through the Fourier pseudo-spectral method(FPSM)and utilizes the online tracking adaptive dynamic programming(OTADP)algorithm to realize real-time trajectory tracking for practical use in the ocean environment.Using the wave prediction technique,the optimal trajectory is generated online through a receding horizon(RH)implementation.A critic neural network(NN)is applied to approximate the optimal cost value function and calculate the error-tracking control by solving the associated Hamilton-Jacobi-Bellman(HJB)equation.The proposed WEC control framework improves computational efficiency and makes the online control feasible in practice.Simulation results show the effects of the receding horizon implementation of FPSM with different window lengths and window functions,while verifying the performances of tracking control and energy absorption of WECs in two different sea conditions.展开更多
The multi-satellite electromagnetic formation flight system is nonlinear and strongly coupled,which makes modeling and optimization challenging.To simplify electromagnetic force evaluation and dynamics modeling,we int...The multi-satellite electromagnetic formation flight system is nonlinear and strongly coupled,which makes modeling and optimization challenging.To simplify electromagnetic force evaluation and dynamics modeling,we introduce a reference frame consistent with each satellite body frame,in which the electromagnetic dipoles and electromagnetic forces are represented as two-dimensional vectors.Then,the maneuver time is divided into time intervals,and different satellite sets are activated in each interval,converting the multi-satellite formation reconfiguration problem into an optimal trajectory problem of each two-satellite subsystem.To this end,a token-based dynamic programming method with a switching penalty of active satellite sets is proposed to determine the sequence of satellite sets participating in each time interval,thereby enabling all satellites to reach their desired states.For the two-satellite subsystem with the objectives of minimizing maneuver time and energy consumption,the Gauss pseudo-spectral method is employed to generate the optimal reconfiguration trajectory.Numerical simulations verify the effectiveness of the proposed optimization method.展开更多
文摘Solar-powered aircraft have attracted great attention owing to their potential for longendurance flight and wide application prospects.Due to the particularity of energy system,flight strategy optimization is a significant way to enhance the flight performance for solar-powered aircraft.In this study,a flight strategy optimization model for high-altitude long-endurance solar-powered aircraft was proposed.This model consists of three-dimensional kinematic model,aerodynamic model,energy collection model,energy store model and energy loss model.To solve the nonlinear optimal control problem with process constraints and terminal constraints,Gauss pseudo-spectral method was employed to discretize the state equations and constraint equations.Then a typical mission flying from given initial point to given final point within a time interval was considered.Results indicate that proper changes of the attitude angle contribute to increasing the energy gained by photovoltaic cells.Utilization of gravitational potential energy can partly take the role of battery pack.Integrating these two measures,the optimized flight strategy can improve the final state of charge compared with current constant-altitude constant-velocity strategy.The optimized strategy brings more profits on condition of lower sunlight intensity and shorter daytime.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.41575025 and 41575024)
文摘To improve the modeling accuracy of radiative transfer,the scattering properties of aerosol particles with irregular shapes and inhomogeneous compositions should be simulated accurately.To this end,a light-scattering model for nonspherical particles is established based on the pseudo-spectral time domain(PSTD)technique.In this model,the perfectly matched layer with auxiliary differential equation(ADE-PML),an excellent absorption boundary condition(ABC)in the finite difference time domain generalized for the PSTD,and the weighted total field/scattered field(TF/SF)technique is employed to introduce the incident light into 3 D computational domain.To improve computational efficiency,the model is further parallelized using the Open MP technique.The modeling accuracy of the PSTD scheme is validated against Lorenz–Mie,Aden–Kerker,T-matrix theory and DDA for spheres,inhomogeneous particles and nonspherical particles,and the influence of the spatial resolution and thickness of ADE-PML on the modeling accuracy is discussed as well.Finally,the parallel computational efficiency of the model is also analyzed.The results show that an excellent agreement is achieved between the results of PSTD and well-tested scattering models,where the simulation errors of extinction efficiencies are generally smaller than 1%,indicating the high accuracy of our model.Despite its low spatial resolution,reliable modeling precision can still be achieved by using the PSTD technique,especially for large particles.To suppress the electromagnetic wave reflected by the absorption layers,a six-layer ADE-PML should be set in the computational domain at least.
基金supported by the Open Fund of Key Laboratory of Geo-detection (China University of Geosciences,Beijing),Ministry of Education (No. GDL0805)
文摘In this article, numerical modeling of borehole radar for well logging in time domain is developed using pseudo-spectral time domain algorithm in axisymmetric cylindrical coordinate for proximate true formation model. The conductivity and relative permittivity logging curves are obtained from the data of borehole radar for well logging. Since the relative permittivity logging curve is not affected by salinity of formation water, borehole radar for well logging has obvious advantages as compared with conventional electrical logging. The borehole radar for well logging is a one-transmitter and two-receiver logging tool. The conductivity and relative permittivity logging curves are obtained successfully by measuring the amplitude radio and the time difference of pulse waveform from two receivers. The calculated conductivity and relative permittivity logging curves are close to the true value of surrounding formation, which tests the usability and reliability of borehole radar for well logging. The numerical modeling of borehole radar for well logging laid the important foundation for researching its logging tool.
文摘Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.
基金Supported by the National"863"Project(No.2014AA06A605)
文摘With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important methods of wave-field simulation. Results of previous studies show that both methods have distinct advantages and disadvantages: Finite difference method has high precision but its dispersion is serious; pseudospectral method considers both computational efficiency and precision but has less precision than finite-difference. The authors consider the complex structural characteristics of the metal ore,furthermore add random media in order to simulate the complex effects produced by metal ore for wave field. First,the study introduced the theories of random media and two forward modelling methods. Second,it compared the simulation results of two methods on fault model. Then the authors established a complex metal ore model,added random media and compared computational efficiency and precision. As a result,it is found that finite difference method is better than pseudo-spectral method in precision and boundary treatment,but the computational efficiency of pseudospectral method is slightly higher than the finite difference method.
文摘In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L<sub>2</sub> and L<sub>∞</sub> error norms are computed to study the accuracy and the simplicity of the presented method.
文摘In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.
文摘This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.
文摘This paper deals with the numerical simulation of incompressible turbulent boundary flow of a flat plate with the pseudo-spectral matrix method. In order to appear more than 10 nodes in the turbulent base-stratum and transition of 43×43 computational grids,a coordinate transformation is put up from physical panel to computational panel. Several zero turbulent models are computed comparatively. The results are credible when comparing with the previous methods.
基金Supported by the Natural Science Foundation of Henan Educational Committee(2003110005)Supported by the Natural Science Foundation of Henan University(XK02069)
文摘In this paper, the pseudo-spectral approximations for a class of the Kdv-Burgers type equation is presented. Convergence and stability of the approximation have been proved by Sobolev's inequalities and the bounded extensive method of the nonlinear function. Finally, the numerical examples are proposed.
基金supported by the National Natural Science Foundation of China(Grant Nos.12171245,11971416,11971242,11901513,11971481)the National Natural Science Foundation of Henan Province(No.222300420280)+3 种基金the Natural Science Foundation of Hunan(Grant Nos.2021JJ40655,2021JJ20053,2023JJ40656)the Program for Innovative Research Team(in Science and Technology)in University of Henan Province(No.23IRTSTHN018)the National Key R&D Program of China(No.2020YFA0709800)the scientific research Fund of Xuchang University(2024ZD010).
文摘In the paper,we consider the coupled nonlinear Schrödinger equation with high degree polynomials in the energy functional that cannot be handled by using the newly proposed quadratic auxiliary variable method.Therefore,we develop the multiple quadratic auxiliary variable approach to deal with coupled systems and construct high-accuracy structure-preserving schemes for the equation.To fix the idea,we first apply the multiple quadratic auxiliary variable approach to the equation and obtain an equivalent system that possesses the original energy and mass.Then,a family of high-accuracy structure-preserving schemes that can conserve the mass and energy is derived by applying the Gauss collocation method and sine pseudo-spectral method to approximate the system in time and space.The given schemes have high-accuracy in time and can both inherit the mass and Hamiltonian energy of the system.Ample numerical results are given to confirm the accuracy and conservation of the developed schemes at last.
基金supported in part by the Natural Science Foundation of Hebei Province(Grant No.A2021205036).
文摘Abstract Recently,the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention.For the nonlinear Schrödinger equation(NLS)with wave operator(NLSW)and weak nonlinearity controlled by a small valueε∈(0,1],an exponential wave integrator Fourier pseudo-spectral(EWIFP)discretization has been developed(Guo et al.,2021)and proved to be uniformly accurate aboutεup to the time atΟ(1/ε^(2))However,the EWIFP method is not time symmetric and can not preserve the discrete energy.As we know,the time symmetry and energy-preservation are the important structural features of the true solution and we hope that this structure can be inherited along the numerical solution.In this work,we propose a time symmetric and energy-preserving exponential wave integrator Fourier pseudo-spectral(SEPEWIFP)method for the NLSW with periodic boundary conditions.Through rigorous error analysis,we establish uniform error bounds of the numerical solution atΟ(h^(mo)+ε^(2-βτ2))up to the time atΟ(1/ε^(β))forβ∈[0,2]where h andτare the mesh size and time step,respectively,and m0 depends on the regularity conditions.The tools for error analysis mainly include cut-off technique and the standard energy method.We also extend the results on error bounds,energy-preservation and time symmetry to the oscillatory NLSW with wavelength atΟ(1/ε^(2))in time which is equivalent to the NLSW with weak nonlinearity.Numerical experiments confirm that the theoretical results in this paper are correct.Our method is novel because that to the best of our knowledge there has not been any energy-preserving exponential wave integrator method for the NLSW.
基金supported by National Natural Science Foundation of China(Grant Nos.10901074,11271171,91130003,11001009 and 11101399)the Province Natural Science Foundation of Jiangxi(Grant No. 20114BAB201011)+2 种基金the Foundation of Department of Education of Jiangxi Province(Grant No.GJJ12174)the State Key Laboratory of Scientific and Engineering Computing,CASsupported by the Youth Growing Foundation of Jiangxi Normal University in 2010
文摘A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-SchrSdinger equations. The scheme is of spectral accuracy in space and of second order in time. The scheme preserves the discrete multisymplectic conservation law and the charge conservation law. Moreover, the residuals of some other conservation laws are derived for the geometric numerical integrator. Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme, and demonstrate the correctness of the theoretical analysis.
文摘Proposes an explicit fully discrete three-level pseudo-spectral scheme with unconditional stability for the Cahn-Hilliard equation. Equations for pseudo-spectral scheme; Analysis of linear stability of critical points.
基金Jialing Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11801277)Tingchun Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11571181)+1 种基金the Natural Science Foundation of Jiangsu Province(Grant No.BK20171454)Qing Lan Project.Yushun Wang’s work is supported by the National Natural Science Foundation of China(Grant Nos.11771213 and 12171245).
文摘This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.
基金supported by the Key R&D Program of Shandong Province,China(No.2021ZLGX04)the Taishan Industrial Experts Programme(No.tsls20231203)。
文摘In this paper,we propose a novel noncausal control framework to address the energy maximization problem of wave energy converters(WECs)subject to constraints.The energy maximization problem of WECs is a constrained optimal control problem.The proposed control framework converts this problem into a reference trajectory tracking problem through the Fourier pseudo-spectral method(FPSM)and utilizes the online tracking adaptive dynamic programming(OTADP)algorithm to realize real-time trajectory tracking for practical use in the ocean environment.Using the wave prediction technique,the optimal trajectory is generated online through a receding horizon(RH)implementation.A critic neural network(NN)is applied to approximate the optimal cost value function and calculate the error-tracking control by solving the associated Hamilton-Jacobi-Bellman(HJB)equation.The proposed WEC control framework improves computational efficiency and makes the online control feasible in practice.Simulation results show the effects of the receding horizon implementation of FPSM with different window lengths and window functions,while verifying the performances of tracking control and energy absorption of WECs in two different sea conditions.
文摘The multi-satellite electromagnetic formation flight system is nonlinear and strongly coupled,which makes modeling and optimization challenging.To simplify electromagnetic force evaluation and dynamics modeling,we introduce a reference frame consistent with each satellite body frame,in which the electromagnetic dipoles and electromagnetic forces are represented as two-dimensional vectors.Then,the maneuver time is divided into time intervals,and different satellite sets are activated in each interval,converting the multi-satellite formation reconfiguration problem into an optimal trajectory problem of each two-satellite subsystem.To this end,a token-based dynamic programming method with a switching penalty of active satellite sets is proposed to determine the sequence of satellite sets participating in each time interval,thereby enabling all satellites to reach their desired states.For the two-satellite subsystem with the objectives of minimizing maneuver time and energy consumption,the Gauss pseudo-spectral method is employed to generate the optimal reconfiguration trajectory.Numerical simulations verify the effectiveness of the proposed optimization method.
