In this work,we are concerned with a time-splitting Fourier pseudospectral(TSFP)discretization for the Klein-Gordon(KG)equation,involving a dimensionless parameterε∈(0,1].In the nonrelativistic limit regime,the smal...In this work,we are concerned with a time-splitting Fourier pseudospectral(TSFP)discretization for the Klein-Gordon(KG)equation,involving a dimensionless parameterε∈(0,1].In the nonrelativistic limit regime,the smallεproduces high oscillations in exact solutions with wavelength of O(ε^(−2))in time.The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time,with both the nonlinear and linear subproblems exactly integrable in time and,respectively,Fourier frequency spaces.The method is fully explicit and time reversible.Moreover,we establish rigorously the optimal error bounds of a second-order TSFP for fixedε=O(1),thanks to an observation that the scheme coincides with a type of trigonometric integrator.As the second task,numerical studies are carried out,with special effortsmade to applying the TSFP in the nonrelativistic limit regime,which are geared towards understanding its temporal resolution capacity and meshing strategy for O(ε^(−2))-oscillatory solutions when 0<ε≪1.It suggests that the method has uniform spectral accuracy in space,and an asymptotic O(ε^(−2)D^(t2))temporal discretization error bound(Dt refers to time step).On the other hand,the temporal error bounds for most trigonometric integrators,such as the well-established Gautschi-type integrator in[6],are O(ε^(−4)D^(t2)).Thus,our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime.These results,either rigorous or numerical,are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.展开更多
This paper introduces an extension of the time-splitting spectral(TSSP)method for solving a general model of three-wave optical interactions,which typically arises from nonlinear optics,when the transmission media has...This paper introduces an extension of the time-splitting spectral(TSSP)method for solving a general model of three-wave optical interactions,which typically arises from nonlinear optics,when the transmission media has competing quadratic and cubic nonlinearities.The key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way,which allows us to develop an explicit and unconditionally stable numerical method for the problem.Furthermore,the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction,is time reversible and time transverse invariant,and conserves the total wave energy(or power or the norm of the solutions)in discretized level.Numerical examples are presented to demonstrate the efficiency and high resolution of the method.Finally the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension(1D)and 2D.展开更多
大地震能够同时激发出许多的地球自由振荡简正模,且地球的椭率、自转和内部的各向异性也会引起简正模的分裂,使各单线态之间的频率更接近(仅为几个μHz),这对地球自由振荡模型的检测提出更高的要求。本文以标准时频变换为基础,推导并验...大地震能够同时激发出许多的地球自由振荡简正模,且地球的椭率、自转和内部的各向异性也会引起简正模的分裂,使各单线态之间的频率更接近(仅为几个μHz),这对地球自由振荡模型的检测提出更高的要求。本文以标准时频变换为基础,推导并验证一种自由振荡模型检测的新方法。以3 S 1模型的检测为例,与经典的FT谱方法和最新的OSE方法相比,该方法具有更高的频率分辨率。展开更多
基金supported by the Singapore A*STAR SERC PSF-Grant 1321202067。
文摘In this work,we are concerned with a time-splitting Fourier pseudospectral(TSFP)discretization for the Klein-Gordon(KG)equation,involving a dimensionless parameterε∈(0,1].In the nonrelativistic limit regime,the smallεproduces high oscillations in exact solutions with wavelength of O(ε^(−2))in time.The key idea behind the TSFP is to apply a time-splitting integrator to an equivalent first-order system in time,with both the nonlinear and linear subproblems exactly integrable in time and,respectively,Fourier frequency spaces.The method is fully explicit and time reversible.Moreover,we establish rigorously the optimal error bounds of a second-order TSFP for fixedε=O(1),thanks to an observation that the scheme coincides with a type of trigonometric integrator.As the second task,numerical studies are carried out,with special effortsmade to applying the TSFP in the nonrelativistic limit regime,which are geared towards understanding its temporal resolution capacity and meshing strategy for O(ε^(−2))-oscillatory solutions when 0<ε≪1.It suggests that the method has uniform spectral accuracy in space,and an asymptotic O(ε^(−2)D^(t2))temporal discretization error bound(Dt refers to time step).On the other hand,the temporal error bounds for most trigonometric integrators,such as the well-established Gautschi-type integrator in[6],are O(ε^(−4)D^(t2)).Thus,our method offers much better approximations than the Gautschi-type integrator in the highly oscillatory regime.These results,either rigorous or numerical,are valid for a splitting scheme applied to the classical relativistic NLS reformulation as well.
基金support from the National University of Singapore grant No.R-146-000-081-112C.Zheng acknowledges the support by National Natural Science Foundation of China(No.10401020)his extended visit at National University of Singapore.
文摘This paper introduces an extension of the time-splitting spectral(TSSP)method for solving a general model of three-wave optical interactions,which typically arises from nonlinear optics,when the transmission media has competing quadratic and cubic nonlinearities.The key idea is to formulate the terms related to quadratic and cubic nonlinearities into a Hermitian matrix in a proper way,which allows us to develop an explicit and unconditionally stable numerical method for the problem.Furthermore,the method is spectral accurate in transverse coordinates and second-order accurate in propagation direction,is time reversible and time transverse invariant,and conserves the total wave energy(or power or the norm of the solutions)in discretized level.Numerical examples are presented to demonstrate the efficiency and high resolution of the method.Finally the method is applied to study dynamics and interactions between three-wave solitons and continuous waves in media with competing quadratic and cubic nonlinearities in one dimension(1D)and 2D.
文摘大地震能够同时激发出许多的地球自由振荡简正模,且地球的椭率、自转和内部的各向异性也会引起简正模的分裂,使各单线态之间的频率更接近(仅为几个μHz),这对地球自由振荡模型的检测提出更高的要求。本文以标准时频变换为基础,推导并验证一种自由振荡模型检测的新方法。以3 S 1模型的检测为例,与经典的FT谱方法和最新的OSE方法相比,该方法具有更高的频率分辨率。
