The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of ma...The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.展开更多
This paper studies a distributed optimal consensus problem for multiple double integrators under bounded velocity and acceleration. Assigned with an in dividual and private convex cost which is dependent on the positi...This paper studies a distributed optimal consensus problem for multiple double integrators under bounded velocity and acceleration. Assigned with an in dividual and private convex cost which is dependent on the position, each age nt n eeds to achieve consensus at the optimum of the aggregate cost under bounded velocity and acceleration. Based on relative positions and velocities to neighbor agents, we design a distributed control law by including the integration feedback of position and velocity errors. By employing quadratic Lyapunov functions, we solve the optimal consensus problem of double-integrators when the fixed topology is strongly connected and weight-balanced. Furthermore, if an initial estimate of the optimum can be known, then control gains can be properly selected to achieve an exponentially fast convergence under bounded velocity and acceleration. The result still holds when the relative velocity is not available, and we also discuss an extension for heterogeneous Euler-Lagrange systems by in verse dynamics control. A nu meric example is provided to illustrate the result.展开更多
Integrator based model is used to describe a wide range of systems in robotics. In this paper, we present an axis-coupled trajectory generation algorithm for chains of integrators with an arbitrary order. Special noti...Integrator based model is used to describe a wide range of systems in robotics. In this paper, we present an axis-coupled trajectory generation algorithm for chains of integrators with an arbitrary order. Special notice has been given to problems with pre-existing nominal plans, which are common in robotic applications. It also handles various type of constraints that can be satisfied on an entire time interval, including non-convex ones which can be transformed into a series of convex constraints through time segmentation. The proposed approach results in a linearly constrained quadratic programming problem, which can be solved effectively with off-the-shelf solvers. A closed-form solution is achievable with only the boundary constraints considered. Finally, the proposed method is tested in real experiments using quadrotors which represent high-order integrator systems.展开更多
In this paper, we propose a variational integrator for nonlinear Schrodinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplect...In this paper, we propose a variational integrator for nonlinear Schrodinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrodinger equations with variable coefficients, cubic nonlinear Schrodinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.展开更多
N-body simulations of the Sun, the planets, and small celestial bodies are frequently used to model the evolution of the Solar System. Large numbers of numerical integrators for performing such simulations have been d...N-body simulations of the Sun, the planets, and small celestial bodies are frequently used to model the evolution of the Solar System. Large numbers of numerical integrators for performing such simulations have been developed and used;see, for example, [1,2]. The primary objective of this paper is to analyse and compare the efficiency and the error growth for different numerical integrators. Throughout the paper, the error growth is examined in terms of the global errors in the positions and velocities, and the relative errors in the energy and angular momentum of the system. We performed numerical experiments for the different integrators applied to the Jovian problem over a long interval of duration, as long as one million years, with the local error tolerance ranging from 10-16 to 10-18.展开更多
ZTE Corporation has signed strategic telecommunications software agreement with two leading providers in Europe and Latin America to optimize its offerings for target customers in
Two new fourth-order three-stage symplectic integrators are specifically designed for a family of Hamiltonian systems,such as the harmonic oscillator,mathematical pendulum and latticeφ4 model.When the nonintegrable l...Two new fourth-order three-stage symplectic integrators are specifically designed for a family of Hamiltonian systems,such as the harmonic oscillator,mathematical pendulum and latticeφ4 model.When the nonintegrable latticeφ4 system is taken as a test model,numerical comparisons show that the new methods have a great advantage over the second-order Verlet symplectic integrators in the accuracy of energy,become explicitly better than the usual non-gradient fourth-order seven-stage symplectic integrator of Forest and Ruth,and are almost equivalent to a fourth-order seven-stage force gradient symplectic integrator of Chin.As the most important advantage,the new integrators are convenient for solving the variational equations of many Hamiltonian systems so as to save a great deal of the computational cost when scanning a lot of orbits for chaos.展开更多
A discrete total variation calculus with variable time steps is presented for mechanico-electrical systems where there exist non-potential and dissipative forces. By using this discrete variation calculus, the symplec...A discrete total variation calculus with variable time steps is presented for mechanico-electrical systems where there exist non-potential and dissipative forces. By using this discrete variation calculus, the symplectic-energy-first integrators for mechanico-electrical systems are derived. To do this, the time step adaptation is employed. The discrete variational principle and the Euler-Lagrange equation are derived for the systems. By using this discrete algorithm it is shown that mechanico-electrical systems are not symplectic and their energies are not conserved unless they are Lagrange mechanico-electrical systems. A practical example is presented to illustrate these results.展开更多
Based on the Magnus integrator method established in linear dynamic systems, an efficiently improved modified Magnus integrator method was proposed for the second-order dynamic systems with time-dependent high frequen...Based on the Magnus integrator method established in linear dynamic systems, an efficiently improved modified Magnus integrator method was proposed for the second-order dynamic systems with time-dependent high frequencies. Firstly, the secondorder dynamic system was reformulated as the first-order system and the frame of reference was transfered by introducing new variables so that highly oscillatory behaviour inherits from the entries in the meantime. Then the modified Magnus integrator method based on local linearization was appropriately designed for solving the above new form and some improved also were presented. Finally, numerical examples show that the proposed methods appear to be quite adequate for integration for highly oscillatory dynamic systems including Hamiltonian systems problem with long time and effectiveness.展开更多
This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton＇s variational principle corresponding to initial value problems is proposed, which is convenient for appli...This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton＇s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and mo- mentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.展开更多
In this work,a consistent and physically accurate implementation of the general framework of unified second-order time accurate integrators via the well-known GSSSS framework in the Discrete Element Method is presente...In this work,a consistent and physically accurate implementation of the general framework of unified second-order time accurate integrators via the well-known GSSSS framework in the Discrete Element Method is presented.The improved tangential displacement evaluation in the present implementation of the discrete element method has been derived and implemented to preserve the consistency of the correct time level evaluation during the time integration process in calculating the algorithmic tangential displacement.Several numerical examples have been used to validate the proposed tangential displacement evaluation;this is in contrast to past practices which only seem to attain the first-order time accuracy due to inconsistent time level implementation with different algorithms for normal and tangential directions.The comparisons with the existing implementation and the superiority of the proposed implementation are given in terms of the convergence rate with improved numerical accuracy in time.Moreover,several schemes via the unified second-order time integrators within the framework of the GSSSS family have been carried out based on the proposed correct implementation.All the numerical results demonstrate that using the existing state-of-the-art implementation reduces the time accuracy to be first-order accurate in time,while the proposed implementation preserves the correct time accuracy to yield second-order.展开更多
For the problem of set point regulation of the liquid level in coupled tank systems, we present a continuous sliding mode control(SMC) with a "conditional integrator", which only provides integral action ins...For the problem of set point regulation of the liquid level in coupled tank systems, we present a continuous sliding mode control(SMC) with a "conditional integrator", which only provides integral action inside the boundary layer. For a special choice of the controller parameters, our design can be viewed as a PID controller with anti-windup and achieves robust regulation.The proposed controller recovers the transient response performance without control chattering. Both full-state feedback as well as output-feedback designs are presented in this work. Our output-feedback design uses a high-gain observer(HGO) which recovers the performance of a state-feedback design where plant parameters are assumed to be known. We consider both interacting as well as non-interacting tanks and analytical results for stability and transient performance are presented in both the cases. The proposed controller continuous SMC with conditional integrators(CSMCCI) provides superior results in terms of the performance measures as well as performance indices than ideal SMC, continuous SMC(CSMC) and continuous SMC with conventional integrator(CSMCI). Experimental results demonstrate good tracking performance in spite of unmodeled dynamics and disturbances.展开更多
In this paper,the authors study the problem of distributed Nash equilibrium seeking of N-player games with high-order integrator dynamics subject to disturbances generated by an uncertain exosystem.Similar problems ha...In this paper,the authors study the problem of distributed Nash equilibrium seeking of N-player games with high-order integrator dynamics subject to disturbances generated by an uncertain exosystem.Similar problems have been studied for disturbances with an exactly known exosystem.Compared with the existing results of high-order integrator dynamics,which can only handle sinusoidal disturbances with known frequencies,this paper aims to handle multi-tone disturbances with unknown frequencies by introducing an adaptive control technique to estimate the unknown frequencies.Technically,when the exosystem is known,the disturbance can be dealt with by the Luenburger observer.In contrast,the Luenburger observer cannot deal with an uncertain exosystem.The authors combine the internal model design and some adaptive control technique to solve the proposed problem.Further,the authors also establish the sufficient condition to guarantee the convergence of the estimated unknown frequencies to the actual values of these frequencies.Two examples are given to verify the proposed algorithm.展开更多
In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an esti...In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.展开更多
We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 i...We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Sehr6dinger equations satisfy trace formulas for the expected mass, energy, and momentum （i. e., linear drifts in these quantities）. Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.展开更多
A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a genera...A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a general formula for composition method.展开更多
Trigonometric integrators for oscillatory linear Hamiltonian differential equations are considered.Under a condition of Hairer&Lubich on the filter functions in the method,a modified energy is derived that is exac...Trigonometric integrators for oscillatory linear Hamiltonian differential equations are considered.Under a condition of Hairer&Lubich on the filter functions in the method,a modified energy is derived that is exactly preserved by trigonometric integrators.This implies and extends a known result on all-time near-conservation of energy.The extension can be applied to linear wave equations.展开更多
We propose Poisson integrators for the numerical integration of separable Poisson systems.We analyze three situations in which Poisson systems are separated in threeways and Poisson integrators can be constructed by u...We propose Poisson integrators for the numerical integration of separable Poisson systems.We analyze three situations in which Poisson systems are separated in threeways and Poisson integrators can be constructed by using the splittingmethod.Numerical results show that the Poisson integrators outperform the higher order non-Poisson integrators in terms of long-termenergy conservation and computational cost.The Poisson integrators are also shown to be more efficient than the canonicalized sympletic methods of the same order.展开更多
Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized al-gorithms and from choosing the best ordering of terms.The c...Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized al-gorithms and from choosing the best ordering of terms.The cost in programming and execution time is minimal,while the performance improvements can be large.In this note we report the influence of term ordering for random systems and for two systems from celestial mechanics that describe particle paths near black holes,quantifying its significance for both optimized and unoptimized methods.We also present a method for the computation of solutions of integrable monomial Hamiltonians that minimizes roundoff error and allows the effective use of compensation summation.展开更多
Multi-organ-on-a-chip(MOOC)technology represents a pivotal direction in the organ-on-a-chip field,seeking to emulate the complex interactions of multiple human organs in vitro through microfluidic systems.This technol...Multi-organ-on-a-chip(MOOC)technology represents a pivotal direction in the organ-on-a-chip field,seeking to emulate the complex interactions of multiple human organs in vitro through microfluidic systems.This technology overcomes the limitations of traditional single-organ models,providing a novel platform for investigating complex disease mechanisms and evaluating drug efficacy and toxicity.Although it demonstrates broad application prospects,its development still faces critical bottlenecks,including inadequate physiological coupling between organs,short functional maintenance durations,and limited real-time monitoring capabilities.Contemporary research is advancing along three key directions,including functional coupling,sensor integration,and full-process automation systems,to propel the technology toward enhanced levels of physiological relevance and predictive accuracy.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos. 10872084 and 10932002)the Research Program of Higher Education of Liaoning Province,China (Grant No. 2008S098)+3 种基金the Program of Supporting Elitists of Higher Education of Liaoning Province,China (Grant No. 2008RC20)the Program of Constructing Liaoning Provincial Key Laboratory,China (Grant No. 2008403009)the Foundation Research Plan of Liaoning educational Bureau,China (Grant No. L2010147)the Youth fund of Liaoning University,China (Grant No. 2008LDQN04)
文摘The variational integrators of autonomous Birkhoff systems are obtained by the discrete variational principle. The geometric structure of the discrete autonomous Birkhoff system is formulated. The discretization of mathematical pendulum shows that the discrete variational method is as effective as symplectic scheme for the autonomous Birkhoff systems.
文摘This paper studies a distributed optimal consensus problem for multiple double integrators under bounded velocity and acceleration. Assigned with an in dividual and private convex cost which is dependent on the position, each age nt n eeds to achieve consensus at the optimum of the aggregate cost under bounded velocity and acceleration. Based on relative positions and velocities to neighbor agents, we design a distributed control law by including the integration feedback of position and velocity errors. By employing quadratic Lyapunov functions, we solve the optimal consensus problem of double-integrators when the fixed topology is strongly connected and weight-balanced. Furthermore, if an initial estimate of the optimum can be known, then control gains can be properly selected to achieve an exponentially fast convergence under bounded velocity and acceleration. The result still holds when the relative velocity is not available, and we also discuss an extension for heterogeneous Euler-Lagrange systems by in verse dynamics control. A nu meric example is provided to illustrate the result.
文摘Integrator based model is used to describe a wide range of systems in robotics. In this paper, we present an axis-coupled trajectory generation algorithm for chains of integrators with an arbitrary order. Special notice has been given to problems with pre-existing nominal plans, which are common in robotic applications. It also handles various type of constraints that can be satisfied on an entire time interval, including non-convex ones which can be transformed into a series of convex constraints through time segmentation. The proposed approach results in a linearly constrained quadratic programming problem, which can be solved effectively with off-the-shelf solvers. A closed-form solution is achievable with only the boundary constraints considered. Finally, the proposed method is tested in real experiments using quadrotors which represent high-order integrator systems.
基金supported by the National Natural Science Foundation of China(Grant No.11401259)the Fundamental Research Funds for the Central Universities,China(Grant No.JUSRR11407)
文摘In this paper, we propose a variational integrator for nonlinear Schrodinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrodinger equations with variable coefficients, cubic nonlinear Schrodinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space.
文摘N-body simulations of the Sun, the planets, and small celestial bodies are frequently used to model the evolution of the Solar System. Large numbers of numerical integrators for performing such simulations have been developed and used;see, for example, [1,2]. The primary objective of this paper is to analyse and compare the efficiency and the error growth for different numerical integrators. Throughout the paper, the error growth is examined in terms of the global errors in the positions and velocities, and the relative errors in the energy and angular momentum of the system. We performed numerical experiments for the different integrators applied to the Jovian problem over a long interval of duration, as long as one million years, with the local error tolerance ranging from 10-16 to 10-18.
文摘ZTE Corporation has signed strategic telecommunications software agreement with two leading providers in Europe and Latin America to optimize its offerings for target customers in
基金by the National Natural Science Foundation of China under Grant No 10873007.
文摘Two new fourth-order three-stage symplectic integrators are specifically designed for a family of Hamiltonian systems,such as the harmonic oscillator,mathematical pendulum and latticeφ4 model.When the nonintegrable latticeφ4 system is taken as a test model,numerical comparisons show that the new methods have a great advantage over the second-order Verlet symplectic integrators in the accuracy of energy,become explicitly better than the usual non-gradient fourth-order seven-stage symplectic integrator of Forest and Ruth,and are almost equivalent to a fourth-order seven-stage force gradient symplectic integrator of Chin.As the most important advantage,the new integrators are convenient for solving the variational equations of many Hamiltonian systems so as to save a great deal of the computational cost when scanning a lot of orbits for chaos.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10672143 and 60575055)the State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciencesthe Natural Science Foundation of Henan Province Government, China (Grant No 0511022200)
文摘A discrete total variation calculus with variable time steps is presented for mechanico-electrical systems where there exist non-potential and dissipative forces. By using this discrete variation calculus, the symplectic-energy-first integrators for mechanico-electrical systems are derived. To do this, the time step adaptation is employed. The discrete variational principle and the Euler-Lagrange equation are derived for the systems. By using this discrete algorithm it is shown that mechanico-electrical systems are not symplectic and their energies are not conserved unless they are Lagrange mechanico-electrical systems. A practical example is presented to illustrate these results.
基金Project supported by the National Natural Science Foundation of China (No.10572119)Program for New Century Excellent Talent of Ministry of Education of China (No.NCET-04-0958)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipmentthe Doctorate Foundation of Northwestern Polytechnical University
文摘Based on the Magnus integrator method established in linear dynamic systems, an efficiently improved modified Magnus integrator method was proposed for the second-order dynamic systems with time-dependent high frequencies. Firstly, the secondorder dynamic system was reformulated as the first-order system and the frame of reference was transfered by introducing new variables so that highly oscillatory behaviour inherits from the entries in the meantime. Then the modified Magnus integrator method based on local linearization was appropriately designed for solving the above new form and some improved also were presented. Finally, numerical examples show that the proposed methods appear to be quite adequate for integration for highly oscillatory dynamic systems including Hamiltonian systems problem with long time and effectiveness.
基金Project supported by the National Natural Science Foundation of China(Nos.11172334 and11202247)the Fundamental Research Funds for the Central Universities(No.2013390003161292)
文摘This paper develops a new approach to construct variational integrators. A simplified unconventional Hamilton＇s variational principle corresponding to initial value problems is proposed, which is convenient for applications. The displacement and mo- mentum are approximated with the same Lagrange interpolation. After the numerical integration and variational operation, the original problems are expressed as algebraic equations with the displacement and momentum at the interpolation points as unknown variables. Some particular variational integrators are derived. An optimal scheme of choosing initial values for the Newton-Raphson method is presented for the nonlinear dynamic system. In addition, specific examples show that the proposed integrators are symplectic when the interpolation point coincides with the numerical integration point, and both are Gaussian quadrature points. Meanwhile, compared with the same order symplectic Runge-Kutta methods, although the accuracy of the two methods is almost the same, the proposed integrators are much simpler and less computationally expensive.
文摘In this work,a consistent and physically accurate implementation of the general framework of unified second-order time accurate integrators via the well-known GSSSS framework in the Discrete Element Method is presented.The improved tangential displacement evaluation in the present implementation of the discrete element method has been derived and implemented to preserve the consistency of the correct time level evaluation during the time integration process in calculating the algorithmic tangential displacement.Several numerical examples have been used to validate the proposed tangential displacement evaluation;this is in contrast to past practices which only seem to attain the first-order time accuracy due to inconsistent time level implementation with different algorithms for normal and tangential directions.The comparisons with the existing implementation and the superiority of the proposed implementation are given in terms of the convergence rate with improved numerical accuracy in time.Moreover,several schemes via the unified second-order time integrators within the framework of the GSSSS family have been carried out based on the proposed correct implementation.All the numerical results demonstrate that using the existing state-of-the-art implementation reduces the time accuracy to be first-order accurate in time,while the proposed implementation preserves the correct time accuracy to yield second-order.
文摘For the problem of set point regulation of the liquid level in coupled tank systems, we present a continuous sliding mode control(SMC) with a "conditional integrator", which only provides integral action inside the boundary layer. For a special choice of the controller parameters, our design can be viewed as a PID controller with anti-windup and achieves robust regulation.The proposed controller recovers the transient response performance without control chattering. Both full-state feedback as well as output-feedback designs are presented in this work. Our output-feedback design uses a high-gain observer(HGO) which recovers the performance of a state-feedback design where plant parameters are assumed to be known. We consider both interacting as well as non-interacting tanks and analytical results for stability and transient performance are presented in both the cases. The proposed controller continuous SMC with conditional integrators(CSMCCI) provides superior results in terms of the performance measures as well as performance indices than ideal SMC, continuous SMC(CSMC) and continuous SMC with conventional integrator(CSMCI). Experimental results demonstrate good tracking performance in spite of unmodeled dynamics and disturbances.
基金supported in part by the Research Grants Council of the Hong Kong Special Administration Region under Grant No.14201420in part by the National Natural Science Foundation of China under GrantNo.61973260.
文摘In this paper,the authors study the problem of distributed Nash equilibrium seeking of N-player games with high-order integrator dynamics subject to disturbances generated by an uncertain exosystem.Similar problems have been studied for disturbances with an exactly known exosystem.Compared with the existing results of high-order integrator dynamics,which can only handle sinusoidal disturbances with known frequencies,this paper aims to handle multi-tone disturbances with unknown frequencies by introducing an adaptive control technique to estimate the unknown frequencies.Technically,when the exosystem is known,the disturbance can be dealt with by the Luenburger observer.In contrast,the Luenburger observer cannot deal with an uncertain exosystem.The authors combine the internal model design and some adaptive control technique to solve the proposed problem.Further,the authors also establish the sufficient condition to guarantee the convergence of the estimated unknown frequencies to the actual values of these frequencies.Two examples are given to verify the proposed algorithm.
基金supported by National Natural Science Foundation of China(Grant No.11671392)
文摘In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov.
文摘We study an explicit exponential scheme for the time discretisation of stochastic SchrS- dinger Equations Driven by additive or Multiplicative It6 Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Sehr6dinger equations satisfy trace formulas for the expected mass, energy, and momentum （i. e., linear drifts in these quantities）. Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.
基金This work is subsidized by the special funds for major state basic research projects (No.1999032800).
文摘A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a general formula for composition method.
文摘Trigonometric integrators for oscillatory linear Hamiltonian differential equations are considered.Under a condition of Hairer&Lubich on the filter functions in the method,a modified energy is derived that is exactly preserved by trigonometric integrators.This implies and extends a known result on all-time near-conservation of energy.The extension can be applied to linear wave equations.
基金supported by the National Natural Science Foundation of China(Grant Nos.11901564 and 12171466).
文摘We propose Poisson integrators for the numerical integration of separable Poisson systems.We analyze three situations in which Poisson systems are separated in threeways and Poisson integrators can be constructed by using the splittingmethod.Numerical results show that the Poisson integrators outperform the higher order non-Poisson integrators in terms of long-termenergy conservation and computational cost.The Poisson integrators are also shown to be more efficient than the canonicalized sympletic methods of the same order.
文摘Many applications in computational physics that use numerical integrators based on splitting and composition can benefit from the development of optimized al-gorithms and from choosing the best ordering of terms.The cost in programming and execution time is minimal,while the performance improvements can be large.In this note we report the influence of term ordering for random systems and for two systems from celestial mechanics that describe particle paths near black holes,quantifying its significance for both optimized and unoptimized methods.We also present a method for the computation of solutions of integrable monomial Hamiltonians that minimizes roundoff error and allows the effective use of compensation summation.
基金supported by the Shenzhen Medical Research Fund(Grant No.A2303049)Guangdong Basic and Applied Basic Research(Grant No.2023A1515010647)+1 种基金National Natural Science Foundation of China(Grant No.22004135)Shenzhen Science and Technology Program(Grant No.RCBS20210706092409020,GXWD20201231165807008,20200824162253002).
文摘Multi-organ-on-a-chip(MOOC)technology represents a pivotal direction in the organ-on-a-chip field,seeking to emulate the complex interactions of multiple human organs in vitro through microfluidic systems.This technology overcomes the limitations of traditional single-organ models,providing a novel platform for investigating complex disease mechanisms and evaluating drug efficacy and toxicity.Although it demonstrates broad application prospects,its development still faces critical bottlenecks,including inadequate physiological coupling between organs,short functional maintenance durations,and limited real-time monitoring capabilities.Contemporary research is advancing along three key directions,including functional coupling,sensor integration,and full-process automation systems,to propel the technology toward enhanced levels of physiological relevance and predictive accuracy.
