MULTISYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHR■DINGER EQUATIONS WITH WAVE OPERATOR

In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.

Multi-Symplectic Fourier Pseudospectral Method for the Kawahara Equation

In this paper,we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiation matrix and discrete Fourier transform.The relationship is crucial for implementing the scheme efficiently.By using the relationship,we can apply the Fast Fourier transform to solve the Kawahara equation.The effectiveness of the proposed methods will be demonstrated by a number of numerical examples.The numerical results also confirm that the global energy and momentum are well preserved.

A New Explicit Symplectic Fourier Pseudospectral Method for Klein-Gordon-Schrodinger Equation

In this paper,we propose an explicit symplectic Fourier pseudospectral method for solving the Klein-Gordon-Schr odinger equation.The key idea is to rewrite the equation as an infinite-dimensional Hamiltonian system and discrete the system by using Fourier pseudospectral method in space and symplectic Euler method in time.After composing two different symplectic Euler methods for the ODEs resulted from semi-discretization in space,we get a new explicit scheme for the target equation which is of second order in space and spectral accuracy in time.The canonical Hamiltonian form of the resulted ODEs is presented and the new derived scheme is proved strictly to be symplectic.The new scheme is totally explicitwhereas symplectic scheme are generally implicit or semi-implicit.Linear stability analysis is carried and a necessary Courant-Friedrichs-Lewy condition is given.The numerical results are reported to test the accuracy and efficiency of the proposed method in long-term computing.

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schrdinger system

We derive a new method for a coupled nonlinear Schr/Sdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove the proposed method preserves the charge and energy conservation laws exactly. In numerical tests, we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions. Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws. These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.

A conservative Fourier pseudospectral algorithm for the nonlinear Schrodinger equation

In this paper, we derive a new method for a nonlinear Schrodinger system by using the square of the first-order Fourier spectral differentiation matrix D1 instead of the traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove that the proposed method preserves the charge and energy conservation laws exactly. A deduction argument is used to prove that the numerical solution is second-order convergent to the exact solutions in ｜｜·｜｜2 norm. Some numerical results are reported to illustrate the efficiency of the new scheme in preserving the charge and energy conservation laws.

Fourier Pseudospectral Solution for a 2D Nonlinear Paraxial Envelope Equation of Laser Interactions in Plasmas

We apply a Fourier pseudospectral algorithm to solve a 2D nonlinear paraxial envelope-equation of laser interactions in plasmas. In this algorithm, we first use the second order Strang time-splitting method to split the envelope-equation into a number of equations, next we spatially discrete the filed quantity and its spatial derivatives in these equations in term of Fourier interpolation polynomials (FFT), finally we sequentially integrate the resultant equations by means of a discrete integration method in order to obtain the solution of the envelope-equation. We carry out several numerical tests to illustrate the efficiency and to determine accuracy of the algorithm. In addition, we conduct a number of numerical experiments to examine its performance. The numerical results have shown that the algorithm is highly efficient and sufficiently accurate to solve the 2D envelope-equation, furthermore, it yields an optimal performance in simulating fundamental phenomena in laser interactions in plasmas.

Explicit multi-symplectic method for the Zakharov-Kuznetsov equation

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.

Multisymplectic implicit and explicit methods for Klein-Gordon-Schrdinger equations

We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrodinger equations.We prove that the implicit method satisfies the charge conservation law exactly.Both methods provide accurate solutions in long-time computations and simulate the soliton collision well.The numerical results show the abilities of the two methods in preserving the charge,energy,and momentum conservation laws.

Conservative method for simulation of a high-order nonlinear Schrdinger equation with a trapped term

We propose a new scheme for simulation of a high-order nonlinear Schrodinger equation with a trapped term by using the mid-point rule and Fourier pseudospectral method to approximate time and space derivatives, respectively. The method is proved to be both charge- and energy-conserved. Various numerical experiments for the equation in different cases are conducted. From the numerical evidence, we see the present method provides an accurate solution and conserves the discrete charge and energy invariants to machine accuracy which are consistent with the theoretical analysis.

Optimal error estimates for Fourier spectral approximation of the generalized KdV equation

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.

Multi-symplectic method for the coupled Schrdinger–KdV equations

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.

Explicit Multi-symplectic Method for a High Order Wave Equation of KdV Type

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.

An Efficient Numerical Method for the Quintic Complex Swift-Hohenberg Equation

In this paper,we present an efficient time-splitting Fourier spectral method for the quintic complex Swift-Hohenberg equation.Using the Strang time-splitting technique,we split the equation into linear part and nonlinear part.The linear part is solved with Fourier Pseudospectral method;the nonlinear part is solved analytically.We show that the method is easy to be applied and second-order in time and spectrally accurate in space.We apply the method to investigate soliton propagation,soliton interaction,and generation of stable moving pulses in one dimension and stable vortex solitons in two dimensions.

Simulation of Inviscid Compressible Flows Using PDE Transform

The solution of systems of hyperbolic conservation laws remains an interesting and challenging task due to the diversity of physical origins and complexity of the physical situations.The present work introduces the use of the partial differential equation(PDE)transform,paired with the Fourier pseudospectral method(FPM),as a new approach for hyperbolic conservation law problems.The PDE transform,based on the scheme of adaptive high order evolution PDEs,has recently been applied to decompose signals,images,surfaces and data to various target functional mode functions such as trend,edge,texture,feature,trait,noise,etc.Like wavelet transform,the PDE transform has controllable time-frequency localization and perfect reconstruction.A fast PDE transform implemented by the fast Fourier Transform(FFT)is introduced to avoid stability constraint of integrating high order PDEs.The parameters of the PDE transform are adaptively computed to optimize the weighted total variation during the time integration of conservation law equations.A variety of standard benchmark problems of hyperbolic conservation laws is employed to systematically validate the performance of the present PDE transform based FPM.The impact of two PDE transform parameters,i.e.,the highest order and the propagation time,is carefully studied to deliver the best effect of suppressing Gibbs’oscillations.The PDE orders of 2-6 are used for hyperbolic conservation laws of low oscillatory solutions,while the PDE orders of 8-12 are often required for problems involving highly oscillatory solutions,such as shock-entropy wave interactions.The present results are compared with those in the literature.It is found that the present approach not only works well for problems that favor low order shock capturing schemes,but also exhibits superb behavior for problems that require the use of high order shock capturing methods.