We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler ...We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.展开更多
The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton ...The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.展开更多
In this paper, the multi-symplectic formulations of the membrane free vibration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-...In this paper, the multi-symplectic formulations of the membrane free vibration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws-a multi-symplectic conservation law （CLS）, a local energy conservation law （ECL） as well as a local momentum conservation law （MCL） --is constructed to discrete the PDEs that are derived from the membrane free vibration equation. The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior,展开更多
This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete mu...This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.展开更多
A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissma...A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.展开更多
In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospect...In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.展开更多
Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical method...Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.展开更多
In this paper, we consider multi-symplectic Fourier pseudospectral method for a high order integrable equation of KdV type, which describes many important physical phenomena. The multi-symplectic structure are constru...In this paper, we consider multi-symplectic Fourier pseudospectral method for a high order integrable equation of KdV type, which describes many important physical phenomena. The multi-symplectic structure are constructed for the equation, and the conservation laws of the continuous equation are presented. The multisymplectic discretization of each formulation is exemplified by the multi-symplectic Fourier pseudospectral scheme. The numerical experiments are given, and the results verify the efficiency of the Fourier pseudospectral method.展开更多
In the present paper, a general solution involv- ing three arbitrary functions for the generalized （2＋1）- dimensional KdV-mKdV equation, which is derived from the generalized （1＋1）-dimensional KdV-mKdV equa- tio...In the present paper, a general solution involv- ing three arbitrary functions for the generalized （2＋1）- dimensional KdV-mKdV equation, which is derived from the generalized （1＋1）-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale （WTC） truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized （2＋1）- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1：o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized （2＋1）- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.展开更多
Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic ...Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.展开更多
Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multisymplectic splitting （MSS） method to solve the two-dimensional nonlinear Schrodinger equation （2D-NLSE） in this pap...Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multisymplectic splitting （MSS） method to solve the two-dimensional nonlinear Schrodinger equation （2D-NLSE） in this paper. It is further shown that the method constructed in this way preserve the global symplectieity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness of the proposed method.展开更多
The mixed state of two-band II-superconductor is analyzed by the multi-symplectic method.As to the Ginzburg-Landau equation depending on time that describes the mixed state of two-band II-superconductor,the multi-symp...The mixed state of two-band II-superconductor is analyzed by the multi-symplectic method.As to the Ginzburg-Landau equation depending on time that describes the mixed state of two-band II-superconductor,the multi-symplectic formulations with several conservation laws:a multi-symplectic conservation law,an energy con-servation law,as well as a momentum conservation law,are presented firstly;then an eighteen points scheme is constructed to simulate the multi-symplectic partial differential equations(PDEs)that are derived from the Ginzburg-Landau equation;finally,based on the simulation results,the volt-ampere characteristic curves are obtained,as well as the relationships between the temperature and resistivity of a suppositional two-band II-superconductor model under different magnetic intensi-ties.From the results of the numerical experiments,it is concluded that the notable property of the mixed state of the two-band II-superconductor is that:The trans-formation temperature decreases and the resistivityρincreases rapidly with the increase of the magnetic intensity B.In addition,the simulation results show that the multi-symplectic method has two remarkable advantages:high accuracy and excellent long-time numerical behavior.展开更多
We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can...We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.展开更多
The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, t...The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.展开更多
In this paper,based on the multi-symplecticity of concatenating symplectic Runge-Kutta-Nystrom(SRKN)methods and symplectic Runge-Kutta-type methods for numerically solving Hamiltonian PDEs,explicit multi-symplectic sc...In this paper,based on the multi-symplecticity of concatenating symplectic Runge-Kutta-Nystrom(SRKN)methods and symplectic Runge-Kutta-type methods for numerically solving Hamiltonian PDEs,explicit multi-symplectic schemes are constructed and investigated,where the nonlinear wave equation is taken as a model problem.Numerical comparisons are made to illustrate the effectiveness of our newly derived explicit multi-symplectic integrators.展开更多
Numerical dispersion relation of the multi-symplectic Runge-Kutta （MSRK） method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the diffe...Numerical dispersion relation of the multi-symplectic Runge-Kutta （MSRK） method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.展开更多
Nonlinear Rossby waves are used to describe typical wave phenomena in large-scale atmosphere andocean.Owing to the nonlinearity of the involved problems,the weakly nonlinear method,ie the derivative ex-pansion method,...Nonlinear Rossby waves are used to describe typical wave phenomena in large-scale atmosphere andocean.Owing to the nonlinearity of the involved problems,the weakly nonlinear method,ie the derivative ex-pansion method,was mainly used to investigate Rossby waves under the combined effects of the generalizedβ-effect and the basic flow effect.The derivative expansion method has the advantage of capturing the multi-scalecharacteristics of wave processes simultaneously.In the case where the perturbation expansion is independentof secular terms,the nonlinear equations describing the amplitude evolution of nonlinear waves were derived,such as the Korteweg-de Vries equation,the Boussinesq equation and Zakharov-Kuznetsov equation.Both quali-tative and quantitative analyses indicate that the generalizedβ-effect is the key factor inducing the evolution ofRossby solitary waves.展开更多
SiC/Al-based composite foams were prepared by a two-step foaming method.The influence of the SiC content and its distribution uniformity on the foaming stability,cell structure,and mechanical properties of the aluminu...SiC/Al-based composite foams were prepared by a two-step foaming method.The influence of the SiC content and its distribution uniformity on the foaming stability,cell structure,and mechanical properties of the aluminum foams was investigated.The macro/micro-features of the aluminum foams were characterized and analyzed.Results demonstrate that an appropriate increase in SiC content and the uniform distribution of SiC can improve the foaming stability,optimize the cell diameter and cell wall thickness,ameliorate the cell distribution,and enhance the hardness and compressive strength of the aluminum foams.However,either insufficient or excessive SiC leads to uneven distribution of SiC particles,which is unfavorable to foaming stability and good cell structure formation.With 6wt%SiC,both the foaming stability and cell structure of the aluminum foam reach the optimal state,resulting in the highest compressive strength and optimal energy absorption capacity.展开更多
In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to ...In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.展开更多
This study presents an effective hybrid simulation approach for simulating broadband ground motion in complex near-fault locations.The approach utilizes a deterministic approach based on the spectral element method(SE...This study presents an effective hybrid simulation approach for simulating broadband ground motion in complex near-fault locations.The approach utilizes a deterministic approach based on the spectral element method(SEM),which is used to simulate low-frequency ground motion(f<1 Hz)by incorporating an innovative efficient discontinuous Galerkin(DG)method for grid division to accurately model basin sedimentary layers at reduced costs.It also introduces a comprehensive hybrid source model for high-frequency random scattering and a nonlinear analysis module for basin sedimentary layers.Deterministic outcomes are combined with modified three-dimensional stochastic finite fault method(3D-EXSIM)simulations of high-frequency ground motion(f>1 Hz).A fourth-order Butterworth filter with zero phase shift is employed for time-domain filtering of low-and high-frequency time series at a crossover frequency of 1 Hz,merging the low and high-frequency ground motions into a broadband time series.Taking an Ms 6.8 Luding earthquake,as an example,this hybrid method was used for a rapid and efficient simulation analysis of broadband ground motion in the region.The accuracy and efficiency of this hybrid method were verified through comparisons with actually observed station data and empirical attenuation curves.Deterministic method simulation results revealed the effects of mountainous topography,basin effects,nonlinear effects within the basin’s sedimentary layers,and a coupling interaction between the basin and the mountains.The findings are consistent with similar studies,showing that near-fault sedimentary basins significantly focus and amplify strong ground motion,and the soil’s nonlinear behavior in the basin influences ground motion to varying extents at different distances from the fault.The mountainous topography impacts the basin’s response to ground motion,leading to barrier effects.This research provides a scientific foundation for seismic zoning,urban planning,and seismic design in nearfault mountain basin regions.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 10971226, 91130013, and 11001270)the National Basic Research Program of China (Grant No. 2009CB723802)
文摘We propose an explicit multi-symplectic method to solve the two-dimensional Zakharov-Kuznetsov equation. The method combines the multi-symplectic Fourier pseudospectral method for spatial discretization and the Euler method for temporal discretization. It is verified that the proposed method has corresponding discrete multi-symplectic conservation laws. Numerical simulations indicate that the proposed scheme is characterized by excellent conservation.
基金Project supported by the National Natural Science Foundation of China (Nos.10572119,10772147,10632030)the Doctoral Program Foundation of Education Ministry of China (No.20070699028)+1 种基金the Natural Science Foundation of Shaanxi Province of China (No.2006A07)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment,Dalian University of Technology.
文摘The generalized Boussinesq equation that represents a group of important nonlinear equations possesses many interesting properties. Multi-symplectic formulations of the generalized Boussinesq equation in the Hamilton space are introduced in this paper. And then an implicit multi-symplectic scheme equivalent to the multi-symplectic Box scheme is constructed to solve the partial differential equations （PDEs） derived from the generalized Boussinesq equation. Finally, the numerical experiments on the soliton solutions of the generalized Boussinesq equation are reported. The results show that the multi-symplectic method is an efficient algorithm with excellent long-time numerical behaviors for nonlinear partial differential equations.
基金the National Natural Science Foundation of China(Nos.10632030 and 10572119)Program for New Century Excellent Talents of Ministry of Education of China(No.NCET-04-0958)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment
文摘In this paper, the multi-symplectic formulations of the membrane free vibration equation with periodic boundary conditions in Hamilton space are considered. The complex method is introduced and a semi-implicit twenty-seven-points scheme with certain discrete conservation laws-a multi-symplectic conservation law （CLS）, a local energy conservation law （ECL） as well as a local momentum conservation law （MCL） --is constructed to discrete the PDEs that are derived from the membrane free vibration equation. The results of the numerical experiments show that the multi-symplectic scheme has excellent long-time numerical behavior,
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10572119, 10772147 and 10632030)the Doctoral Program Foundation of Education Ministry of China (Grant No 20070699028)+1 种基金the National Natural Science Foundation of Shaanxi Province of China (Grant No 2006A07)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment
文摘This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.
基金Project supported by the Program for Innovative Research Team in Science and Technology in Fujian Province University,China,the Quanzhou High Level Talents Support Plan,China(Grant No.2017ZT012)the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University,China(Grant No.ZQN-YX502)
文摘A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.
基金Project supported by the National Natural Science Foundation of China(Grant No.91130013)the Open Foundation of State Key Laboratory of High Performance Computing
文摘In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.
文摘Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.
文摘In this paper, we consider multi-symplectic Fourier pseudospectral method for a high order integrable equation of KdV type, which describes many important physical phenomena. The multi-symplectic structure are constructed for the equation, and the conservation laws of the continuous equation are presented. The multisymplectic discretization of each formulation is exemplified by the multi-symplectic Fourier pseudospectral scheme. The numerical experiments are given, and the results verify the efficiency of the Fourier pseudospectral method.
基金supported by the National Natural Science Foundation of China (11002115,10972182,11172239)the Science Foundation of Aviation of China (2010ZB53021)+5 种基金the China Postdoctoral Science Special Foundation (201003682)111 project(B07050) to the Northwestern Polytechnical Universitythe NPU Foundation for Fundamental Research (JC200938,JC20110259)the Doctoral Program Foundation of Education Ministry of China(20106102110019)the Open Foundation of State Key Laboratory of Mechanical System & Vibration (MSV-2011-21)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802)
文摘In the present paper, a general solution involv- ing three arbitrary functions for the generalized （2＋1）- dimensional KdV-mKdV equation, which is derived from the generalized （1＋1）-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale （WTC） truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized （2＋1）- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1：o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized （2＋1）- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.
基金supported by the National Natural Science Foundation of China (Nos. 10772147 and10632030)the Ph. D. Program Foundation of Ministry of Education of China (No. 20070699028)+2 种基金the Natural Science Foundation of Shaanxi Province of China (No. 2006A07)the Open Foundationof State Key Laboratory of Structural Analysis of Industrial Equipment (No. GZ0802)the Foundation for Fundamental Research of Northwestern Polytechnical University
文摘Nonlinear wave equations have been extensively investigated in the last sev- eral decades. The Landau-Ginzburg-Higgs equation, a typical nonlinear wave equation, is studied in this paper based on the multi-symplectic theory in the Hamilton space. The multi-symplectic Runge-Kutta method is reviewed, and a semi-implicit scheme with certain discrete conservation laws is constructed to solve the first-order partial differential equations （PDEs） derived from the Landau-Ginzburg-Higgs equation. The numerical re- sults for the soliton solution of the Landau-Ginzburg-Higgs equation are reported, showing that the multi-symplectic Runge-Kutta method is an efficient algorithm with excellent long-time numerical behaviors.
基金Supported by the Natural Science Foundation of China under Grant No.0971226the 973 Project of China under Grant No.2009CB723802+1 种基金the Research Innovation Fund of Hunan Province under Grant No.CX2011B011the Innovation Fund of NUDT under Grant No.B110205
文摘Using the idea of splitting numerical methods and the multi-symplectic methods, we propose a multisymplectic splitting （MSS） method to solve the two-dimensional nonlinear Schrodinger equation （2D-NLSE） in this paper. It is further shown that the method constructed in this way preserve the global symplectieity exactly. Numerical experiments for the plane wave solution and singular solution of the 2D-NLSE show the accuracy and effectiveness of the proposed method.
基金the National Natural Science Foundation of China(Grant Nos.10572119,10772147 and 10632030)the Doctoral Program Founda-tion of Education Ministry of China(Grant No.20070699028)+1 种基金the Natural Science Foundation of Shaanxi Province of China(Grant No.2006A07)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment
文摘The mixed state of two-band II-superconductor is analyzed by the multi-symplectic method.As to the Ginzburg-Landau equation depending on time that describes the mixed state of two-band II-superconductor,the multi-symplectic formulations with several conservation laws:a multi-symplectic conservation law,an energy con-servation law,as well as a momentum conservation law,are presented firstly;then an eighteen points scheme is constructed to simulate the multi-symplectic partial differential equations(PDEs)that are derived from the Ginzburg-Landau equation;finally,based on the simulation results,the volt-ampere characteristic curves are obtained,as well as the relationships between the temperature and resistivity of a suppositional two-band II-superconductor model under different magnetic intensi-ties.From the results of the numerical experiments,it is concluded that the notable property of the mixed state of the two-band II-superconductor is that:The trans-formation temperature decreases and the resistivityρincreases rapidly with the increase of the magnetic intensity B.In addition,the simulation results show that the multi-symplectic method has two remarkable advantages:high accuracy and excellent long-time numerical behavior.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10971226,91130013,and 11001270)the National Basic Research Program of China(Grant No.2009CB723802)+1 种基金the Research Innovation Fund of Hunan Province,China (Grant No.CX2011B011)the Innovation Fund of National University of Defense Technology,China(Grant No.B120205)
文摘We propose a multi-symplectic wavelet splitting equations. Based on its mu]ti-symplectic formulation, method to solve the strongly coupled nonlinear SchrSdinger the strongly coupled nonlinear SchrSdinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570the Open Foundation of State Key Laboratory of High Performance Computing of China+1 种基金the Research Fund of the National University of Defense Technology under Grant No JC15-02-02the Fund from HPCL
文摘The two-component Camassa–Holm equation includes many intriguing phenomena. We propose a multi-symplectic compact method to solve the two-component Camassa–Holm equation. Based on its multi-symplectic formulation, the proposed method is derived by the sixth-order compact finite difference method in spatial discretization and the symplectic implicit midpoint scheme in temporal discretization. Numerical experiments finely describe the velocity and density variables in the two-component integrable system and distinctly display the evolvement of the singular solutions. Moreover, the proposed method shows good conservative properties during long-time numerical simulation.
基金supported by the Director Innovation Foundation of ICMSEC and AMSS,the Foundation of CAS,the NNSFC(No.19971089 and No.10371128)the National Basic Research Program of China under the Grant 2005CB321701.
文摘In this paper,based on the multi-symplecticity of concatenating symplectic Runge-Kutta-Nystrom(SRKN)methods and symplectic Runge-Kutta-type methods for numerically solving Hamiltonian PDEs,explicit multi-symplectic schemes are constructed and investigated,where the nonlinear wave equation is taken as a model problem.Numerical comparisons are made to illustrate the effectiveness of our newly derived explicit multi-symplectic integrators.
基金The NNSF (10471054) of Chinathe China Postdoctoral Science Foundationthe Youth Foundation of Institute of Mathematics, Jilin University
文摘Numerical dispersion relation of the multi-symplectic Runge-Kutta （MSRK） method for linear Hamiltonian PDEs is derived in the present paper, which is shown to be a discrete counterpart to that possessed by the differential equation. This provides further understanding of MSRK methods. However, much still remains to be investigated further.
文摘Nonlinear Rossby waves are used to describe typical wave phenomena in large-scale atmosphere andocean.Owing to the nonlinearity of the involved problems,the weakly nonlinear method,ie the derivative ex-pansion method,was mainly used to investigate Rossby waves under the combined effects of the generalizedβ-effect and the basic flow effect.The derivative expansion method has the advantage of capturing the multi-scalecharacteristics of wave processes simultaneously.In the case where the perturbation expansion is independentof secular terms,the nonlinear equations describing the amplitude evolution of nonlinear waves were derived,such as the Korteweg-de Vries equation,the Boussinesq equation and Zakharov-Kuznetsov equation.Both quali-tative and quantitative analyses indicate that the generalizedβ-effect is the key factor inducing the evolution ofRossby solitary waves.
基金Doctoral Startup Fund(20192066,20212028)Laijin Excellent Doctoral Fund(20202021)+1 种基金Scientific and Technological Innovation of Colleges and Universities in Shanxi Province(2020L0342)Fundamental Research Program of Shanxi Province(202303021222178)。
文摘SiC/Al-based composite foams were prepared by a two-step foaming method.The influence of the SiC content and its distribution uniformity on the foaming stability,cell structure,and mechanical properties of the aluminum foams was investigated.The macro/micro-features of the aluminum foams were characterized and analyzed.Results demonstrate that an appropriate increase in SiC content and the uniform distribution of SiC can improve the foaming stability,optimize the cell diameter and cell wall thickness,ameliorate the cell distribution,and enhance the hardness and compressive strength of the aluminum foams.However,either insufficient or excessive SiC leads to uneven distribution of SiC particles,which is unfavorable to foaming stability and good cell structure formation.With 6wt%SiC,both the foaming stability and cell structure of the aluminum foam reach the optimal state,resulting in the highest compressive strength and optimal energy absorption capacity.
基金Supported in part by Natural Science Foundation of Guangxi(2023GXNSFAA026246)in part by the Central Government's Guide to Local Science and Technology Development Fund(GuikeZY23055044)in part by the National Natural Science Foundation of China(62363003)。
文摘In this paper,we consider the maximal positive definite solution of the nonlinear matrix equation.By using the idea of Algorithm 2.1 in ZHANG(2013),a new inversion-free method with a stepsize parameter is proposed to obtain the maximal positive definite solution of nonlinear matrix equation X+A^(*)X|^(-α)A=Q with the case 0<α≤1.Based on this method,a new iterative algorithm is developed,and its convergence proof is given.Finally,two numerical examples are provided to show the effectiveness of the proposed method.
基金National Natural Science Foundation of China under Grant Nos.U2139208 and 52278516Key Laboratory of Earthquake Engineering and Engineering Vibration,China Earthquake Administration under Grant No.2024D15Key Laboratory of Soft Soil Characteristic and Engineering Environment,Tianjin Chengjian University under Grant No.2022SCEEKL003。
文摘This study presents an effective hybrid simulation approach for simulating broadband ground motion in complex near-fault locations.The approach utilizes a deterministic approach based on the spectral element method(SEM),which is used to simulate low-frequency ground motion(f<1 Hz)by incorporating an innovative efficient discontinuous Galerkin(DG)method for grid division to accurately model basin sedimentary layers at reduced costs.It also introduces a comprehensive hybrid source model for high-frequency random scattering and a nonlinear analysis module for basin sedimentary layers.Deterministic outcomes are combined with modified three-dimensional stochastic finite fault method(3D-EXSIM)simulations of high-frequency ground motion(f>1 Hz).A fourth-order Butterworth filter with zero phase shift is employed for time-domain filtering of low-and high-frequency time series at a crossover frequency of 1 Hz,merging the low and high-frequency ground motions into a broadband time series.Taking an Ms 6.8 Luding earthquake,as an example,this hybrid method was used for a rapid and efficient simulation analysis of broadband ground motion in the region.The accuracy and efficiency of this hybrid method were verified through comparisons with actually observed station data and empirical attenuation curves.Deterministic method simulation results revealed the effects of mountainous topography,basin effects,nonlinear effects within the basin’s sedimentary layers,and a coupling interaction between the basin and the mountains.The findings are consistent with similar studies,showing that near-fault sedimentary basins significantly focus and amplify strong ground motion,and the soil’s nonlinear behavior in the basin influences ground motion to varying extents at different distances from the fault.The mountainous topography impacts the basin’s response to ground motion,leading to barrier effects.This research provides a scientific foundation for seismic zoning,urban planning,and seismic design in nearfault mountain basin regions.
