Multisymplectic Scheme Research Articles

Multi-Symplectic Method for the Two-Component Camassa–Holm (2CH) System

In this paper, the multi-symplectic formulations of the two-component Camassa–Holm system are presented. Both the multi-symplectic structure and two local conservation laws of the generalized two-component Camassa–Holm model are proposed for its first-order canonical form. Then, combining the Fourier pseudo-spectral method in the spatial domain with the midpoint method in the time dimension, the multi-symplectic Fourier pseudo-spectral scheme is constructed for the first-order canonical form. Meanwhile, the discrete scheme of the residuals of the multi-symplectic structure and two local conservation laws are also provided. By using the multi-symplectic Fourier pseudo-spectral scheme, the evolution of one- and two-soliton solutions for the generalized two-component Camassa–Holm model is regained. The structure-preserving properties and the reliability of the numerical scheme are illustrated by the tiny numerical residuals (less than 3.5 × 10−8) of the conservation laws as well as the tiny numerical variations (less than 1 × 10−9) of the amplitudes and the propagating velocities of the solitons.

Dynamic analysis on an asymmetric spatial dumbbell-type model

The structural symmetric breaking of the spatial structure will enhance the coupling dynamic behaviors and bring new challenges for the dynamic analysis on the spatial structure inevitably. The main contribution of this paper is proposing a structure-preserving iteration method to investigate the effects of the dynamic symmetry-breaking factors on the dynamic behaviors of the rigid-flexible coupling systems. Firstly, the spatial flexible damping beam assembled with two particles on both ends is simplified as an asymmetric spatial dumbbell-type model. For this model, the coupling dynamic equations are presented based on the Hamiltonian variational principle. Then, connecting the symplectic precise integration method for the ordinary differential equations mainly controlling the plane motion of the model and the generalized multi-symplectic scheme for the partial differential equation mainly controlling the transverse vibration of the beam, a structure-preserving numerical iteration method is proposed, which provides a new way to analysis the dynamic behaviors of the coupling problems with the symmetric breaking. Finally, the effects of the absolute mass and the mass ratio of the additional particles on the orbit radius, the orbit true anomaly velocity, the attitude angle velocity of the model and the transverse vibration of the flexible beam are reproduced by using the structure-preserving numerical iteration method in the numerical simulations. Particularly, the effects of the additional particles on the attitude stability of the model and the vibration dissipation of the flexible damping beam are investigated, which gives some guidance for the attitude adjustment strategy and the vibration control method for the spatial flexible structure with the structural symmetric breaking.

Multi-Symplectic Simulation on Soliton-Collision for Nonlinear Perturbed Schrödinger Equation

Seeking solitary wave solutions and revealing their interactional characteristics for nonlinear evolution equations help us lot to comprehend the motion laws of the microparticles. As a local nonlinear dynamic behavior, the soliton-collision is difficult to be reproduced numerically. In this paper, the soliton-collision process in the nonlinear perturbed Schrödinger equation is simulated employing the multi-symplectic method. The multi-symplectic formulations are derived including the multi-symplectic form and three local conservation laws of the nonlinear perturbed Schrödinger equation. Employing the implicit midpoint rule, we construct a multi-symplectic scheme, which is equivalent to the Preissmann box scheme, for the nonlinear perturbed Schrödinger equation. The elegant structure-preserving properties of the multi-symplectic scheme are illustrated by the tiny maximum absolute residual of the discrete multi-symplectic structure at each time step in the numerical simulations. The effects of the perturbation strength on the soliton-collision in the nonlinear perturbed Schrödinger equation are reported in the numerical results in detail.

Multi-symplectic simulations of W/M-shape-peaks solitons and cuspons for FORQ equation

The FORQ equation, which is bi-Hamiltonian, integrable, and hosts of a range of soliton solutions, is viewed afresh from the viewpoint of multi-symplectic structures. A multi-symplectic formulation is derived and used as a guide in the construction of numerical methods for simulation; in particular, a multi-symplectic Preissmann box scheme is constructed for the FORQ equation, which is second-order accurate in both space and time. Simulations show that the cuspons and the W/M-shape-peaks solitons for the FORQ equation are reproduced accurately. To verify the precision and the validity of the numerical results, the relative errors between the numerical results and the theoretical solutions, and the maximum values of the errors of the local conservation laws are presented.

Meshless symplectic and multi-symplectic scheme for the coupled nonlinear Schrödinger system based on local RBF approximation

Meshless symplectic and multi-symplectic scheme for the coupled nonlinear Schrödinger system based on local RBF approximation

Conformal structure‐preserving method for two‐dimensional damped nonlinear fractional Schrödinger equation

AbstractThis paper mainly analyzes conservation laws and convergence of splitting conformal multisymplectic scheme for solving the two‐dimensional damped nonlinear fractional Schrödinger equation. In order to obtain conformal multisymplectic scheme, first using Strang splitting method, the original problem is split into conservative multisymplectic and dissipative multisymplectic systems. The conservative multisymplectic system is numerically solved and the dissipative part is solved exactly. And then there shows the corresponding conservation laws and numerical scheme, in which the implicit midpoint method is used in time and the Fourier pseudospectral method is used in space, it also preserves conformal multisymplectic, the global conformal symplectic and global conformal mass conservation laws. Most important of all, we discuss convergence of the proposed scheme which is second‐order accuracy in time and spectral accuracy in space. Finally, the validity and accuracy of the theoretical results are verified by several numerical examples.

A meshless multi-symplectic local radial basis function collocation scheme for the “good” Boussinesq equation

A novel meshless multi-symplectic scheme is proposed for the “good” Boussinesq equation. The scheme consists of a local radial basis function (RBF) collocation method (LRBFCS) in space and a symplectic integrator in time. The LRBFCS is simple and efficient, since only a sparse banded linear system has to be solved. Moreover, it can avoid the ill-conditioned problem and shape-parameter-sensitivity of the global RBF method. The multi-symplectic LRBFCS (MLRBFCS) is more accurate than traditional methods. Numerical experiments with uniform knots and random knots are designed to verify the effectiveness of the method.

Exponentially fitted multisymplectic scheme for conservative Maxwell equations with oscillary solutions.

Aiming at conservative Maxwell equations with periodic oscillatory solutions, we adopt exponentially fitted trapezoidal scheme to approximate the temporal and spatial derivatives. The scheme is a multisymplectic scheme. Under periodic boundary condition, the scheme satisfies two discrete energy conservation laws. The scheme also preserves two discrete divergences. To reduce computation cost, we split the original Maxwell equations into three local one-dimension (LOD) Maxwell equations. Then exponentially fitted trapezoidal scheme, applied to the resulted LOD equations, generates LOD multisymplectic scheme. We prove the unconditional stability and convergence of the LOD multisymplectic scheme. Convergence of numerical dispersion relation is also analyzed. At last, we present two numerical examples with periodic oscillatory solutions to confirm the theoretical analysis. Numerical results indicate that the LOD multisymplectic scheme is efficient, stable and conservative in solving conservative Maxwell equations with oscillatory solutions. In addition, to one-dimension Maxwell equations, we apply least square method and LOD multisymplectic scheme to fit the electric permittivity by using exact solution disturbed with small random errors as measured data. Numerical results of parameter inversion fit well with measured data, which shows that least square method combined with LOD multisymplectic scheme is efficient to estimate the model parameter under small random disturbance.

On the Derivation of Multisymplectic Variational Integrators for Hyperbolic PDEs Using Exponential Functions

We investigated the derivation of numerical methods for solving partial differential equations, focusing on those that preserve physical properties of Hamiltonian systems. The formulation of these properties via symplectic forms gives rise to multisymplectic variational schemes. By using analogy with the smooth case, we defined a discrete Lagrangian density through the use of exponential functions, and derived its Hamiltonian by Legendre transform. This led to a discrete Hamiltonian system, the symplectic forms of which obey the conservation laws. The integration schemes derived in this work were tested on hyperbolic-type PDEs, such as the linear wave equations and the non-linear seismic wave equations, and were assessed for their accuracy and the effectiveness by comparing them with those of standard multisymplectic ones. Our error analysis and the convergence plots show significant improvements over the standard schemes.

Classical and Multisymplectic Schemes for Linearized KdV Equation: Numerical Results and Dispersion Analysis

We construct three finite difference methods to solve a linearized Korteweg–de-Vries (KdV) equation with advective and dispersive terms and specified initial and boundary conditions. Two numerical experiments are considered; case 1 is when the coefficient of advection is greater than the coefficient of dispersion, while case 2 is when the coefficient of dispersion is greater than the coefficient of advection. The three finite difference methods constructed include classical, multisymplectic and a modified explicit scheme. We obtain the stability region and study the consistency and dispersion properties of the various finite difference methods for the two cases. This is one of the rare papers that analyse dispersive properties of methods for dispersive partial differential equations. The performance of the schemes are gauged over short and long propagation times. Absolute and relative errors are computed at a given time at the spatial nodes used.

Generalized Multi-Symplectic Numerical Implementation of Dynamic Responses for Saturated Poroelastic Timoshenko Beam

Based on the porous media theory and Timoshenko beam theory, properties of dynamic responses in fluid-solid coupled incompressible saturated poroelastic Timoshenko beam are investigated by generalized multi-symplectic method. Dynamic response equation set of incompressible saturated poroelastic Timoshenko beam is presented at first. Then a first order generalized multi-symplectic form of this dynamic response equation set is constructed, and errors of generalized multi-symplectic conservation law, generalized multi-symplectic local momentum and generalized multi-symplectic local energy are also derived. A Preissmann Box generalized multi-symplectic scheme of the dynamic response equation set is presented, the discrete errors of generalized multi-symplectic conservation law, generalized multi-symplectic local momentum conservation law and generalized multi-symplectic local energy conservation law are also obtained. In view of the dynamic responses of incompressible saturated poroelastic Timoshenko cantilever beam with two ends permeable and free end subjected to the step load, the transverse dynamic response process of the solid skeleton is simulated numerically, the evolution processes of solid effective stress and the equivalent moment of the pore fluid pressure over time are also presented numerically. The effects of fluid-solid coupled interaction parameter and slenderness ratio of the beam on the solid dynamic response process are revealed, as well as the effects on all generalized multi-symplectic numerical errors are checked simultaneously. From results obtained, the processes for solid deflection, solid effective stress and the equivalent moment of the pore fluid pressure approaching to their steady response values are all shortened with increasing of fluid-solid coupled interaction parameter, while the response process of solid deflection and the pore fluid equivalent moment are lengthened with increasing of slenderness ratio of the beam. Moreover, the steady value of solid deflection is much closer to the static deflection value of classic single phase elastic Euler-Bernoulli beam with increasing of the slenderness ratio. As time goes on, the solid skeleton of the beam will support all outside load, so equivalent moment of the pore fluid pressure becomes zero at last. In addition, it is presented all generalized multi-symplectic numerical errors decrease with the decreasing of parameters representing the dissipation effect for the dynamic system.

Multisymplectic structure-preserving scheme for the coupled Gross–Pitaevskii equations

In this paper, we study numerically the dynamics of the rotating Bose–Einstein condensates (BECs) modelled by the coupled Gross–Pitaevskii (CGP) equations with angular momentum rotating terms. First, the multisymplectic structure of the CGP equations is investigated by introducing some canonical momenta which allows us to construct the corresponding multisymplectic schemes. Then applying the midpoint rule in both temporal and spatial directions, a multisymplectic scheme is proposed for the CGP equations. The conservative properties and the convergence analysis are discussed for the multisymplectic scheme. The cut-off function technique is utilized for the error estimation. Numerical examples are carried out to verify the conservative property and the convergence rate.

Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics.

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

Dispersion Analysis of Multi-symplectic Scheme for the Nonlinear Schrödinger Equations

In this paper, we study the dispersive properties of multi-symplectic discretizations for the nonlinear Schrodinger equations. The numerical dispersion relation and group velocity are investigated. It is found that the numerical dispersion relation is relevant when resolving the nonlinear Schrodinger equations.

Multi-symplectic integrator of the generalized KdV-type equation based on the variational principle

The variational principle is used to construct a multi-symplectic structure of the generalized KdV-type equation. Accordingly, the local energy conservation law, the local momentum conservation law, and the Cartan form of the generalized KdV-type equation are given. An explicit multi-symplectic scheme for the generalized KdV equation based on the Fourier pseudo-spectral method and the symplectic Euler scheme is constructed. Through a numerical examination, the explicit multi-symplectic Fourier pseudo-spectral scheme for the generalized KdV equation not only preserve the discrete global energy conservation law and the global momentum conservation law with high accuracy, but show long-time numerical stability as well.

A Review on Stochastic Multi-symplectic Methods for Stochastic Maxwell Equations

Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al. (A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255–268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.

Second order conformal multi-symplectic method for the damped Korteweg–de Vries equation**Project supported by the Program for Innovative Research Team in Science and Technology in Fujian Province University, China, the Quanzhou High Level Talents Support Plan, China (Grant No. 2017ZT012), and the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University, China (Grant No. ZQN-YX502).

A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries (DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme, we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations (PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior.

High Order Compact Multisymplectic Scheme for Coupled Nonlinear Schrödinger-KdV Equations

High Order Compact Multisymplectic Scheme for Coupled Nonlinear Schrödinger-KdV Equations

Efficient high-order structure-preserving methods for the generalized Rosenau-type equation with power law nonlinearity

Based on the multi-symplectic Hamiltonian formula of the generalized Rosenau-type equation, a multi-symplectic scheme and an energy-preserving scheme are proposed. To improve the accuracy of the solution, we apply the composition technique to the obtained schemes to develop high-order schemes which are also multi-symplectic and energy-preserving respectively. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results verify that all the proposed schemes have satisfactory performance in providing accurate solution and preserving the discrete mass and energy invariants. Numerical results also show that although each basic time step is divided into several composition steps, the computational efficiency of the composition schemes is much higher than that of the non-composite schemes.

Numerical analysis of two local conservative methods for two-dimensional nonlinear Schrödinger equation

In this paper, we propose two local conservative methods for solving two-dimensional nonlinearSchrodinger equation. Without consideration of the boundary conditions, they can preserve corresponding local energy and momentum conservation laws exactly at arbitrary spatial-temporal regions. Meanwhile, the charge, global energy and global momentum conservation laws can be conserved under suitable boundary conditions.Numerical analysis, including conservative properties and error estimation analysis, are investigated.Furthermore, a similar multi-symplectic Preissman scheme is constructed as comparison.Numerical experiments show the advantages of the proposed methods during long-time numerical simulations and validate the analysis.