In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations,and develop a stochastic multisymplectic method for numerically solving a kind of stochastic nonlinear Schrodinger equations.It is shown that the stochasticmulti-symplecticmethod preserves themultisymplectic structure,the discrete charge conservation law,and deduces the recurrence relation of the discrete energy.Numerical experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision.
This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations.We investigate the dissipativity properties of (k,l)algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid.The finitedimensional and infinite-dimensional dissipativity results of (k,l)-algebraically stable Runge-Kutta methods are obtained.
A multisymplectic Fourier pseudo-spectral scheme,which exactly preserves the discrete multisymplectic conservation law,is presented to solve the Klein-Gordon-Schrdinger equations.The scheme is of spectral accuracy in space and of second order in time.The scheme preserves the discrete multisymplectic conservation law and the charge conservation law.Moreover,the residuals of some other conservation laws are derived for the geometric numerical integrator.Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme,and demonstrate the correctness of the theoretical analysis.
In this paper,we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrodinger equations.We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law,discrete charge conservation law and discrete energy evolution law almost surely.Numerical experiments confirm well the theoretical analysis results.Furthermore,we present a detailed numerical investigation of the optical phenomena based on the compact scheme.By numerical experiments for various amplitudes of noise,we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time.In particular,if the noise is relatively strong,the soliton will be totally destroyed.Meanwhile,we observe that the phase shift is sensibly modified by the noise.Moreover,the numerical results present inelastic interaction which is different from the deterministic case.
We study the construction of symplectic Runge-Kutta methods for stochastic Hamiltonian systems(SHS).Three types of systems,SHS with multiplicative noise,special separable Hamiltonians and multiple additive noise,respectively,are considered in this paper.Stochastic Runge-Kutta(SRK)methods for these systems are investigated,and the corresponding conditions for SRK methods to preserve the symplectic property are given.Based on the weak/strong order and symplectic conditions,some effective schemes are derived.In particular,using the algebraic computation,we obtained two classes of high weak order symplectic Runge-Kutta methods for SHS with a single multiplicative noise,and two classes of high strong order symplectic Runge-Kutta methods for SHS with multiple multiplicative and additive noise,respectively.The numerical case studies confirm that the symplectic methods are efficient computational tools for long-term simulations.
The local one-dimensional multisymplectic scheme(LOD-MS)is developed for the three-dimensional(3D)Gross-Pitaevskii(GP)equation in Bose-Einstein condensates.The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional(LOD)method.The 3D GP equation is split into three linear LOD Schrodinger equations and an exactly solvable nonlinear Hamiltonian ODE.The three linear LOD Schrodinger equations are multisymplectic which can be approximated by multisymplectic integrator(MI).The conservative properties of the proposed scheme are investigated.It is masspreserving.Surprisingly,the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable.This is impossible for conventional MIs in nonlinear Hamiltonian context.The numerical results show that the LOD-MS can simulate the original problems very well.They are consistent with the numerical analysis.
Nonlinear dynamical systems are sometimes under the influence of random fluctuations. It is desirable to examine possible bifurcations for stochastic dynamical systems when a parameter varies. A computational analysis is conducted to investigate bifurcations of a simple dynamical system under non-Gaussian α-stable Lévy motions, by examining the changes in stationary probability density functions for the solution orbits of this stochastic system. The stationary probability density functions are obtained by solving a nonlocal Fokker-Planck equation numerically. This allows numerically investigating phenomenological bifurcation, or P-bifurcation, for stochastic differential equations with non-Gaussian Lévy noises.
In this work,we concern with the numerical comparison between different kinds of design points in least square(LS)approach on polynomial spaces.Such a topic is motivated by uncertainty quantification(UQ).Three kinds of design points are considered,which are the Sparse Grid(SG)points,the Monte Carlo(MC)points and the Quasi Monte Carlo(QMC)points.We focus on three aspects during the comparison:(i)the convergence properties;(ii)the stability,i.e.the properties of the resulting condition number of the design matrix;(iii)the robustness when numerical noises are present in function values.Several classical high dimensional functions together with a random ODE model are tested.It is shown numerically that(i)neither the MC sampling nor the QMC sampling introduce the low convergence rate,namely,the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used;(ii)The use of SG points admits better convergence properties only for very low dimensional problems(say d≤2);(iii)The QMC points,being deterministic,seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view.
