On Symplectic and Multisymplectic Schemes for the KdV Equation

We examine some symplectic and multisymplectic methods for the notorious Korteweg–de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discretizations, several interesting schemes are constructed and studied. Our essential conclusions are that it is possible to design very stable, conservative difference schemes for the nonlinear, conservative KdV equation. Among the best of such schemes are methods which are symplectic or multisymplectic. Semi-explicit, symplectic schemes can be very effective in many situations. Compact box schemes are effective in ensuring that no artificial wiggles appear in the approximate solution. A family of box schemes is constructed, of which the multisymplectic box scheme is a prominent member, which are particularly stable on coarse space–time grids     

Local discontinuous Galerkin methods based on the multisymplectic formulation for two kinds of Hamiltonian PDEs

ABSTRACT

This paper examines the novel local discontinuous Galerkin (LDG) discretization for Hamiltonian PDEs based on its multisymplectic formulation. This new kind of LDG discretizations possess one major advantage over other standard LDG method, which, through specially chosen numerical fluxes, states the preservation of discrete conservation laws (i.e. energy), and also the multisymplectic structure while the symplectic time integration is adopted. Moreover, the corresponding local multisymplectic conservation law holds at the units of elements instead of each node. Taking the nonlinear Schrödinger equation and the KdV equation as the examples, we illustrate the derivations of discrete conservation laws and the corresponding numerical fluxes. Numerical experiments by using the modified LDG method are demonstrated for the sake of validating our theoretical results.     

Sixth-order symplectic and symmetric explicit ERKN schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations

This paper is devoted to the analysis of the sixth-order symplectic and symmetric explicit extended Runge–Kutta–Nyström (ERKN) schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations. Fourteen practical sixth-order symplectic and symmetric explicit ERKN schemes are constructed, and their phase properties are investigated. The paper is accompanied by five numerical experiments, including a nonlinear two-dimensional wave equation. The numerical results in comparison with the sixth-order symplectic and symmetric Runge–Kutta–Nyström methods and a Gautschi-type method demonstrate the efficiency and robustness of the new explicit schemes for solving multi-frequency oscillatory nonlinear Hamiltonian equations.     

Derivation of the multisymplectic Crank–Nicolson scheme for the nonlinear Schrödinger equation

The Crank–Nicolson scheme as well as its modified schemes is widely used in numerical simulations for the nonlinear Schrödinger equation. In this paper, we prove the multisymplecticity and symplecticity of this scheme. Firstly, we reconstruct the scheme by the concatenating method and present the corresponding discrete multisymplectic conservation law. Based on the discrete variational principle, we derive a new variational integrator which is equivalent to the Crank–Nicolson scheme. Therefore, we prove the multisymplecticity again from the Lagrangian framework. Symplecticity comes from the proper discrete Hamiltonian structure and symplectic integration in time. We also analyze this scheme on stability and convergence including the discrete mass conservation law. Numerical experiments are presented to verify the efficiency and invariant-preserving property of this scheme. Comparisons with the multisymplectic Preissmann scheme are made to show the superiority of this scheme.     

Dispersion,group velocity,and multisymplectic discretizations

This paper examines the dispersive properties of multisymplectic discretizations of linear and nonlinear PDEs. We focus on a leapfrog in space and time scheme and the Preissman box scheme. We find that the numerical dispersion relations are monotonic and determine the relationship between the group velocities of the different numerical schemes. The group velocity dispersion is used to explain the qualitative differences in the numerical solutions obtained with the different schemes. Furthermore, the numerical dispersion relation is found to be relevant when determining the ability of the discretizations to resolve nonlinear dynamics.     

A symplectic algorithm for wave equations

Numerical schemes for finite-dimensional Hamiltonian system which preserve the symplectic structure are generalized to infinite-dimensional Hamiltonian systems and applied to construct finite difference schemes for the nonlinear wave equation. The numerical results show that these schemes compare favorably with conventional difference methods. Furthermore, the successful long-term tracking capability for these Hamiltonian schemes is remarkable and striking.     

三类保辛摄动及其数值比较

摄动法近似应当保辛．本文指出,有限元位移法自动保辛,有限元混合能表示也保辛．摄动法的刚度阵Taylor级数展开能证明保辛;混合能的Taylor级数展开摄动也证明了保辛．但传递辛矩阵的Taylor级数展开摄动却不能保辛．辛矩阵只能在乘法群下保辛,故传递辛矩阵的保辛摄动必须采用正则变换的乘法．虽然刚度阵加法摄动、混合能矩阵加法摄动与传递辛矩阵正则变换乘法摄动都保辛,但这3种摄动近似并不相同．最后通过数值例题给出了对比．     

Poisson schemes for Hamiltonian systems on Poisson manifolds

In this paper, we construct Poisson difference schemes of any order accuracy based on Padé approximation for linear Hamiltonian systems on Poisson manifolds with constant coefficients. For nonlinear Hamiltonian systems on Poisson manifolds, we point out that symplectic diagonal implicit Runge-Kutta methods are also Poisson schemes. The preservation of distinguished functions and quadratic first integrals of the original Hamiltonian systems of these schemes are also discussed.     

Multisymplectic numerical method for the regularized long-wave equation

In this paper, we derive a 6-point multisymplectic Preissman scheme for the regularized long-wave equation from its Bridges' multisymplectic form. Backward error analysis is implemented for the new scheme. The performance and the efficiency of the new scheme are illustrated by solving several test examples. The obtained results are presented and compared with previous methods. Numerical results indicate that the new multisymplectic scheme can not only obtain satisfied solutions, but also keep three invariants of motion very well.     

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.     

Symplectic integration schemes for the ABC flow

Explicit symplectic integration schemes for the Arnold-Beltrami-Childress flows are presented and compared to a fourth order Runge-Kutta method. For moderate accuracy the symplectic schemes are more efficient for the calculation of stable orbits. The structure of the Hamiltonian prevents the implementation of symplectic methods with constant time steps.     

A semi-explicit multi-symplectic splitting scheme for a 3-coupled nonlinear Schrödinger equation

In this paper, we mainly propose an efficient semi-explicit multi-symplectic splitting scheme to solve a 3-coupled nonlinear Schrödinger (3-CNLS) equation. Based on its multi-symplectic formulation, the 3-CNLS equation can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic Fourier pseudospectral method and symplectic Euler method are employed in spatial and temporal discretizations, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical experiments for the unstable plane waves show the effectiveness of the proposed method during long-time numerical calculation.     

Symplectic integrators for discrete nonlinear Schrödinger systems

Symplectic methods for integrating canonical and non-canonical Hamiltonian systems are examined. A general form for higher order symplectic schemes is developed for non-canonical Hamiltonian systems using generating functions and is directly applied to the Ablowitz–Ladik discrete nonlinear Schrödinger system. The implicit midpoint scheme, which is symplectic for canonical systems, is applied to a standard Hamiltonian discretization. The symplectic integrators are compared with an explicit Runge–Kutta scheme of the same order. The relative performance of the integrators as the dimension of the system is varied is discussed.     

二维对流扩散方程基于三角形网格的特征差分格式

引言 对流扩散方程描述了众多的物理现象,其数值算法研究一直受到重视11一叩3一‘4].在这方面,特征差分方法和特征有限元方法是非常有效的两种方法[1一6}.特征差分方法计算简单,但适应区域不够灵活.特征有限元方法则对矩形区域或一般区域均可得到令人满意的结果,但特征有限     

小参数摄动法与保辛

应用数学与力学经常使用小参数摄动近似．在物理与力学中有大量保守体系的分析．保守体系的特点是保辛．本文指出小参数摄动法保辛的问题应予考虑．位移法摄动是保辛的,而辛矩阵的加法摄动则未能保辛．数值例题给出了对比．     

大阻尼杆振动的广义多辛算法

本文基于Bridges教授建立的多辛算法理论及其Hamilton变分原理,采用广义多辛算法研究了大阻尼杆的阻尼振动特性.引入正交动量后,首先将描述大阻尼杆振动的控制方程降阶为一阶Hamilton近似对称形式,即广义多辛形式;随后采用中点离散方法构造形式广义多辛形式的中点Box广义多辛离散格式;最后通过计算机模拟研究大阻尼杆振动过程中的耗散效应.研究结果表明,本文构造的广义多辛算法不仅能够保持系统守恒型几何性质,同时能够再现系统的耗散效应.     

High-order closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems

In this paper we investigate the connection between closed Newton-Cotes, trigonometrically-fitted differential methods and symplectic integrators. From the literature we can see that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. The well-known open Newton-Cotes differential methods presented as multilayer symplectic integrators by Zhu et al. [W. Zhu, X. Zhao, Y. Tang, Journal of Chem. Phys. 104 (1996) 2275]. The construction of multistep symplectic integrators based on the open Newton-Cotes integration methods is investigated by Chiou and Wu [J.C. Chiou, S.D. Wu, J. Chem. Phys. 107 (1997) 6894]. In this paper we investigate the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. After this we construct trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration proceeds.     

Locally Conservative Continuous Galerkin FEM for Pressure Equation in Two-Phase Flow Model in Subsurfaces

A typical two-phase model for subsurface flow couples the Darcy equation for pressure and a transport equation for saturation in a nonlinear manner. In this paper, we study a combined method consisting of continuous Galerkin finite element methods (CGFEMs) followed by a post-processing technique for Darcy equation and a nodal centered finite volume method (FVM) with upwind schemes for the saturation transport equation, in which the coupled nonlinear problem is solved in the framework of operator decomposition. The post-processing technique is applied to CGFEM solutions to obtain locally conservative fluxes which ensures accuracy and robustness of the FVM solver for the saturation transport equation. We applied both upwind scheme and upwind scheme with slope limiter for FVM on triangular meshes in order to eliminate the non-physical oscillations. Various numerical examples are presented to demonstrate the performance of the overall methodology.     

High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations

In this paper, we develop a new kind of multisymplectic integrator for the coupled nonlinear Schrödinger (CNLS) equations. The CNLS equations are cast into multisymplectic formulation. Then it is split into a linear multisymplectic formulation and a nonlinear Hamiltonian system. The space of the linear subproblem is approximated by a high-order compact (HOC) method which is new in multisymplectic context. The nonlinear subproblem is integrated exactly. For splitting and approximation, we utilize an HOC–SMS integrator. Its stability and conservation laws are investigated in theory. Numerical results are presented to demonstrate the accuracy, conservation laws, and to simulate various solitons as well, for the HOC–SMS integrator. They are consistent with our theoretical analysis.     

Meshless Conservative Scheme for Multivariate Nonlinear Hamiltonian PDEs

For multivariate nonlinear Hamiltonian equations, we propose a meshless conservative method by using radial basis approximation. Based on the method of lines, we first discretize the Hamiltonian functional using radial basis function interpolation, and then obtain a finite-dimensional semi-discrete Hamiltonian system. Moreover, we define a discrete symplectic form and verify that it is an approximation to the continuous one and is conserved with respect to time. For time discretization, two conservative methods (symplectic method and energy-conserving method) are employed to derive the full-discretized system. Approximation errors together with conservation properties including symplecticity and energy are discussed in detail. Finally, we present several numerical examples to illustrate that our method is accurate and effective when processing nonlinear Hamiltonian equations with scattered nodes. Besides, the numerical results also confirm the excellent conservation properties of the proposed method.