In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability...In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.展开更多
In this pape,~ we study uniform L1-stability and asymptotic completeness of the Vlasov-Yukawa-Boltzmann （V-Y-B） system. For a sufficiently small rand smooth initial data, we show that classical solutions exist globa...In this pape,~ we study uniform L1-stability and asymptotic completeness of the Vlasov-Yukawa-Boltzmann （V-Y-B） system. For a sufficiently small rand smooth initial data, we show that classical solutions exist globally and satisfy dispersion estimates, uniform L1-stability with respect to initial data and scattering type estimate. We show that the short range nature of interactions due to the Yukawa potential enables us to construct robust Lyapunov type functional to derive scattering states. In the zero mass limit of force carrier particles, we also show that the classical solutions to the V-Y-B system converge to that of the Vlasov-Poisson-Boltzmann （V-P-B） system in any finite time interval展开更多
The paper deals with the g2-stability analysis of multi-input-multi-output （MIMO） systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monoton...The paper deals with the g2-stability analysis of multi-input-multi-output （MIMO） systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monotone and quasi-monotone nonlin- earities. For nonlinear MIMO systems that are described by differential equations, most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates. In contrast, a non-Lyapunov framework is employed here to derive new and more general g2-stability conditions in the frequency domain. These conditions have the following features： i） They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block, ii） For certain cases of the periodic time-varying gain, they contain, depending on the multiplier function chosen, no restrictions on the normalized rate of variation of the time-varying gain, but, for other periodic/aperiodic time-varying gains, they do. Overall, even when specialized to periodic-coefficient linear and nonlinear MIMO systems, the stability conditions are distinct from and less restrictive than recent results in the literature. No comparable results exist in the literature for aperiodic time-varying gains. Furthermore, some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented. Examples are given to illustrate a few of the stability theorems.展开更多
New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F...New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F（z）, in negative feedback with a matrix of periodic/aperiodic gains A（k）, k = 0,1, 2,... and a vector of certain classes of non-monotone/monotone nonlinearities φp（-）, without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features： i） They involve the positive definiteness of the real part （as evaluated on ｜z｜ = 1） of the product of Г （z） and a matrix multiplier function of z. ii） For periodic A（k）, one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A（k）, but for aperiodic A（k）, which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of （A（k ＋ 1）,A（k））, k = 1, 2 iii） They are distinct from and less restrictive than recent results in the literature.展开更多
An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our sc...An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When polynomial of degree at most p is used for spatial approximation,our scheme is verified to converge at a rate of O(τ^(2)+h^(p)+1/2).Numerical results in both 2D and 3D are provided to validate our theoretical prediction.展开更多
A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain met...A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain method(FETD),in our scheme,discontinuous Galerkinmethods are used to discretize not only the spatial domain but also the temporal domain.The proposed numerical scheme is proved to be unconditionally stable,and a convergent rate O((△t)^(r+1)+h^(k+1/2))is established under the L^(2)-normwhen polynomials of degree atmost r and k are used for temporal and spatial approximation,respectively.Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction.An ultra-convergence of order(△t)^(2r+1) in time step is observed numerically for the numerical fluxes w.r.t.temporal variable at the grid points.展开更多
基金supported by NSFC(11341002)NSFC(11171104,10871066)+1 种基金the Construct Program of the Key Discipline in Hunansupported in part by US National Science Foundation under Grant DMS-1115530
文摘In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the un-derlying problem. Unconditional L2-stability and error estimate of order O？τr＋1＋hk＋1/2？ are obtained when polynomials of degree at most r and k are used for the temporal dis-cretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r＋1 in temporal variable t.
基金partially supported by a National Research Foundation of Korea Grant funded by the Korean Government(2014R1A2A205002096)supported by BK21 Plus-KAIST
文摘In this pape,~ we study uniform L1-stability and asymptotic completeness of the Vlasov-Yukawa-Boltzmann （V-Y-B） system. For a sufficiently small rand smooth initial data, we show that classical solutions exist globally and satisfy dispersion estimates, uniform L1-stability with respect to initial data and scattering type estimate. We show that the short range nature of interactions due to the Yukawa potential enables us to construct robust Lyapunov type functional to derive scattering states. In the zero mass limit of force carrier particles, we also show that the classical solutions to the V-Y-B system converge to that of the Vlasov-Poisson-Boltzmann （V-P-B） system in any finite time interval
文摘The paper deals with the g2-stability analysis of multi-input-multi-output （MIMO） systems, governed by integral equations, with a matrix of periodic/aperiodic time-varying gains and a vector of monotone, non-monotone and quasi-monotone nonlin- earities. For nonlinear MIMO systems that are described by differential equations, most of the literature on stability is based on an application of quadratic forms as Lyapunov-function candidates. In contrast, a non-Lyapunov framework is employed here to derive new and more general g2-stability conditions in the frequency domain. These conditions have the following features： i） They are expressed in terms of the positive definiteness of the real part of matrices involving the transfer function of the linear time-invariant block and a matrix multiplier function that incorporates the minimax properties of the time-varying linear/nonlinear block, ii） For certain cases of the periodic time-varying gain, they contain, depending on the multiplier function chosen, no restrictions on the normalized rate of variation of the time-varying gain, but, for other periodic/aperiodic time-varying gains, they do. Overall, even when specialized to periodic-coefficient linear and nonlinear MIMO systems, the stability conditions are distinct from and less restrictive than recent results in the literature. No comparable results exist in the literature for aperiodic time-varying gains. Furthermore, some new stability results concerning the dwell-time problem and time-varying gain switching in linear and nonlinear MIMO systems with periodic/aperiodic matrix gains are also presented. Examples are given to illustrate a few of the stability theorems.
文摘New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F（z）, in negative feedback with a matrix of periodic/aperiodic gains A（k）, k = 0,1, 2,... and a vector of certain classes of non-monotone/monotone nonlinearities φp（-）, without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features： i） They involve the positive definiteness of the real part （as evaluated on ｜z｜ = 1） of the product of Г （z） and a matrix multiplier function of z. ii） For periodic A（k）, one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A（k）, but for aperiodic A（k）, which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of （A（k ＋ 1）,A（k））, k = 1, 2 iii） They are distinct from and less restrictive than recent results in the literature.
基金supported by the National Natural Science Foundation of China(Grant Nos.11171104,91430107)the Construct Program of the Key Discipline in Hunan.This first author is supported by Hunan Provincial Innovation Foundation for Postgraduate under Grant CX2013B217.
文摘An implicit discontinuous Galerkin method is introduced to solve the timedomain Maxwell’s equations in metamaterials.The Maxwell’s equations in metamaterials are represented by integral-differential equations.Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain.The fully discrete numerical scheme is proved to be unconditionally stable.When polynomial of degree at most p is used for spatial approximation,our scheme is verified to converge at a rate of O(τ^(2)+h^(p)+1/2).Numerical results in both 2D and 3D are provided to validate our theoretical prediction.
基金supported by the NSFC(11171104 and 10871066)the Science and Technology Grant of Guizhou Province(LKS[2010]05)+2 种基金supported by the NSFC(11171104 and 10871066)Hunan Provincial Innovation Foundation for Postgraduate(#CX2010B211).supported by the US National Science Foundation through grant DMS-1115530the Ministry of Education of China through the Changjiang Scholars program,the Guangdong Provincial Government of China through the”Computational Science Innovative Research Team”program,and Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University.
文摘A fully discrete discontinuous Galerkin method is introduced for solving time-dependent Maxwell’s equations.Distinguished from the Runge-Kutta discontinuous Galerkin method(RKDG)and the finite element time domain method(FETD),in our scheme,discontinuous Galerkinmethods are used to discretize not only the spatial domain but also the temporal domain.The proposed numerical scheme is proved to be unconditionally stable,and a convergent rate O((△t)^(r+1)+h^(k+1/2))is established under the L^(2)-normwhen polynomials of degree atmost r and k are used for temporal and spatial approximation,respectively.Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction.An ultra-convergence of order(△t)^(2r+1) in time step is observed numerically for the numerical fluxes w.r.t.temporal variable at the grid points.
