A LINEARLY-IMPLICIT STRUCTURE-PRESERVING EXPONENTIAL TIME DIFFERENCING SCHEME FOR HAMILTONIAN PDEs

In the paper,we propose a novel linearly implicit structure-preserving algorithm,which is derived by combing the invariant energy quadratization approach with the exponential time differencing method,to construct efficient and accurate time discretization scheme for a large class of Hamiltonian partial differential equations(PDEs).The proposed scheme is a linear system,and can be solved more efficient than the original energy-preserving ex-ponential integrator scheme which usually needs nonlinear iterations.Various experiments are performed to verify the conservation,efficiency and good performance at relatively large time step in long time computations.

Neumann边界条件下sine-Gordon方程的高效保能量算法

本文对Neumann边界条件下的sine-Gordon方程提出两类新的全离散高效保能量算法.首先考虑在两种不同空间网格上应用cosine拟谱方法去发展空间保结构格式,导出两个有限维Hamilton常微分方程系统.然后,将预估校正型的Crank-Nicolson格式和投影方法相结合,得到一类全离散保能量算法.另外,本文对sine-Gordon方程引入一个补充变量,将原始模型转化成一个松弛系统,这使得保结构算法更容易被发展.本文针对等价的松弛系统仍采用cosine拟谱方法和预估校正的CrankNicolson格式进行离散,发展了另一类新的保能量算法.本文提出的数值格式不仅保持系统的原始能量,而且可以通过离散cosine变换进行高效快速求解.最后,数值实验验证了格式的数值精度、计算效率和优秀性态.

A SIXTH ORDER AVERAGED VECTOR FIELD METHOD

In this paper, based on the theory of rooted trees and B-series, we propose the concrete formulas of the substitution law for the trees of order = 5. With the help of the new substitution law, we derive a B-series integrator extending the averaged vector field （AVF） methods for general Hamiltonian system to higher order. The new integrator turns out to be order of six and exactly preserves energy for Hamiltonian systems. Numerical exper- iments are presented to demonstrate the accuracy and the energy-preserving property of the sixth order AVF method.

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.

HIGH ORDER COMPACT MULTISYMPLECTIC SCHEME FOR COUPLED NONLINEAR SCHRODINGER-KDV EQUATIONS

In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing some canonical momenta. To simulate the problem efficiently, the CNLS-KdV equations are approximated by a high order compact method in space which preserves N semi-discrete multisymplectic conservation laws. We then discretize the semi-discrete system by using a symplectic midpoint scheme in time. Thus, a full-discrete multisymplectic scheme is obtained for the CNLS-KdV equations. The conservation laws of the full-discrete scheme are analyzed. Some numerical experiments are presented to further verify the convergence and conservation laws of the new scheme.

Arbitrarily High-Order Energy-Preserving Schemes for the Camassa-Holm Equation Based on the Quadratic Auxiliary Variable Approach

In this paper,we present a quadratic auxiliary variable(QAV)technique to develop a novel class of arbitrarily high-order energy-preserving algorithms for the Camassa-Holm equation.The QAV approach is first utilized to transform the original equation into a reformulated QAV system with a consistent initial condition.Then the reformulated QAV system is discretized by applying the Fourier pseudo-spectral method in space and the symplectic Runge-Kutta methods in time,which arrives at a class of fully discrete schemes.Under the consistent initial condition,they can be rewritten as a new fully discrete system by eliminating the introduced auxiliary variable,which is rigorously proved to be energy-preserving and symmetric.Ample numerical experiments are conducted to confirm the expected order of accuracy,conservative property and efficiency of the proposed methods.The presented numerical strategy makes it possible to directly apply a special class of Runge-Kutta methods to develop energy-preserving algorithms for a general conservative system with any polynomial energy.

GeometricNumerical Integration for Peakon b-Family Equations

In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.

THE STRUCTURE-PRESERVING METHODS FOR THE DEGASPERIS-PROCESI EQUATION

This paper gives several structure-preserving schemes for the Degasperis-Procesi equation which has bi-Hamiltonian structures consisted of both complex and non-local Hamiltonian differential operators. For this sake, few structure-preserving schemes have been proposed so far. In our work, based on one of the bi-Hamiltonian structures, a symplectic scheme and two new energy-preserving schemes are constructed. The symplecticity comes straightly from the application of the implicit midpoint method on the semi-discrete system which is proved to remain Hamiltonian, while the energy conservation is derived by the combination of the averaged vector field method of second and fourth order, respectively. Some numerical results are presented to show that the three schemes do have the advantages in numerical stability, accuracy in long time computing and ability to preserve the invariants of the DP equation.

TWO NOVEL CLASSES OF ARBITRARY HIGH-ORDER STRUCTURE-PRESERVING ALGORITHMS FOR CANONICAL HAMILTONIAN SYSTEMS

In this paper,we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems.The one class is the symplectic scheme,which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method,respectively.Each member in these schemes is symplectic for any fixed parameter.A more general form of generating functions is introduced,which generalizes the three classical generating functions that are widely used to construct symplectic algorithms.The other class is a novel family of energy and quadratic invariants preserving schemes,which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step.The existence of the solutions of these schemes is verified.Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.

A New Framework of Convergence Analysis for Solving the General Nonlinear Schrodinger Equation using the Fourier Pseudo-Spectral Method in Two Dimensions

This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.

A LINEARLY-IMPLICIT ENERGY-PRESERVING ALGORITHM FOR THE TWO-DIMENSIONAL SPACE-FRACTIONAL NONLINEAR SCHRÖDINGER EQUATION BASED ON THE SAV APPROACH

The main objective of this paper is to present an efficient structure-preserving scheme,which is based on the idea of the scalar auxiliary variable approach,for solving the twodimensional space-fractional nonlinear Schrodinger equation.First,we reformulate the equation as an canonical Hamiltonian system,and obtain a new equivalent system via introducing a scalar variable.Then,we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction.After that,applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version.As expected,the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step.Finally,numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.

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 Conformal Energy-Conserved Method for Maxwell’s Equations with Perfectly Matched Layers

In this paper,a conformal energy-conserved scheme is proposed for solving the Maxwell’s equations with the perfectly matched layer.The equations are split as a Hamiltonian system and a dissipative system,respectively.The Hamiltonian system is solved by an energy-conserved method and the dissipative system is integrated exactly.With the aid of the Strang splitting,a fully-discretized scheme is obtained.The resulting scheme can preserve the five discrete conformal energy conservation laws and the discrete conformal symplectic conservation law.Based on the energy method,an optimal error estimate of the scheme is established in discrete L2-norm.Some numerical experiments are addressed to verify our theoretical analysis.