Multisymplectic Formulation Research Articles

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

AbstractIn this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

A multi-symplectic numerical integrator for the two-component Camassa–Holm equation

A new multi-symplectic formulation of the two-component Camassa-Holm equation (2CH) is presented, and the associated local conservation laws are shown to correspond to certain well-known Hamiltonian functionals. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. Furthermore, this scheme preserves exactly two discrete versions of the Casimir functions of 2CH. Numerical experiments show that the proposed numerical scheme has good conservation properties.

Multi-symplectic method for the coupled Schrödinger—KdV equations

In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrödinger-KdV equations (CSKE) 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.

Multi-symplectic magnetohydrodynamics

AbstractA multi-symplectic formulation of ideal magnetohydrodynamics (MHD) is developed based on the Clebsch variable variational principle in which the Lagrangian consists of the kinetic minus the potential energy of the MHD fluid modified by constraints using Lagrange multipliers that ensure mass conservation, entropy advection with the flow, the Lin constraint, and Faraday's equation (i.e. the magnetic flux is Lie dragged with the flow). The analysis is also carried out using the magnetic vector potentialÃwhere α=Ã⋅dxis Lie dragged with the flow, andB=∇×Ã. The multi-symplectic conservation laws give rise to the Eulerian momentum and energy conservation laws. The symplecticity or structural conservation laws for the multi-symplectic system corresponds to the conservation of phase space. It corresponds to taking derivatives of the momentum and energy conservation laws and combining them to producen(n−1)/2 extra conservation laws, wherenis the number of independent variables. Noether's theorem for the multi-symplectic MHD system is derived, including the case of non-Cartesian space coordinates, where the metric plays a role in the equations.

Ion acoustic traveling waves

AbstractModels for traveling waves in multi-fluid plasmas give essential insight into fully nonlinear wave structures in plasmas, not readily available from either numerical simulations or from weakly nonlinear wave theories. We illustrate these ideas using one of the simplest models of an electron–proton multi-fluid plasma for the case where there is no magnetic field or a constant normal magnetic field present. We show that the traveling waves can be reduced to a single first-order differential equation governing the dynamics. We also show that the equations admit a multi-symplectic Hamiltonian formulation in which both the space and time variables can act as the evolution variable. An integral equation useful for calculating adiabatic, electrostatic solitary wave signatures for multi-fluid plasmas with arbitrary mass ratios is presented. The integral equation arises naturally from a fluid dynamics approach for a two fluid plasma, with a given mass ratio of the two species (e.g. the plasma could be an electron–proton or an electron–positron plasma). Besides its intrinsic interest, the integral equation solution provides a useful analytical test for numerical codes that include a proton–electron mass ratio as a fundamental constant, such as for particle in cell (PIC) codes. The integral equation is used to delineate the physical characteristics of ion acoustic traveling waves consisting of hot electron and cold proton fluids.

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.

MULTISYMPLECTIC AND POLYSYMPLECTIC STRUCTURES ON FIBER BUNDLES

We propose new definitions of the concepts of a multisymplectic structure and of a polysymplectic structure which extend previous ones so as to cover the cases that are of interest in mathematical physics: they are tailored to apply to fiber bundles, rather than just manifolds, and at the same time they are sufficiently specific to allow us to prove Darboux theorems for the existence of canonical local coordinates. A key role is played by the notion of "symbol" of a multisymplectic form, which is a polysymplectic form representing its leading order contribution, thus clarifying the relation between these two closely related but not identical concepts.

Generating functionals and Lagrangian partial differential equations

The main goal of this paper is to derive an alternative characterization of the multisymplectic form formula for classical field theories using the geometry of the space of boundary values. We review the concept of Type-I/II generating functionals defined on the space of boundary data of a Lagrangian field theory. On the Lagrangian side, we define an analogue of Jacobi's solution to the Hamilton–Jacobi equation for field theories, and we show that by taking variational derivatives of this functional, we obtain an isotropic submanifold of the space of Cauchy data, described by the so-called multisymplectic form formula. As an example of the latter, we show that Lorentz's reciprocity principle in electromagnetism is a particular instance of the multisymplectic form formula. We also define a Hamiltonian analogue of Jacobi's solution, and we show that this functional is a Type-II generating functional. We finish the paper by defining a similar framework of generating functions for discrete field theories, and we show that for the linear wave equation, we recover the multisymplectic conservation law of Bridges.

Multi-symplectic scheme for the coupled Schrödinger—Boussinesq equations

In this paper, a multi-symplectic Hamiltonian formulation is presented for the coupled Schrödinger—Boussinesq equations (CSBE). Then, a multi-symplectic scheme of the CSBE is derived. The discrete conservation laws of the Langmuir plasmon number and total perturbed number density are also proved. Numerical experiments show that the multi-symplectic scheme simulates the solitary waves for a long time, and preserves the conservation laws well.

Geometric structures associated with a simple Cartan 3-form

We introduce the notion of a manifold admitting a simple compact Cartan 3-form ω3. We study algebraic types of such manifolds, in particular those having skew-symmetric torsion, or those associated with a closed or coclosed 3-form ω3. We prove the existence of an algebra of multi-symplectic forms ϕl on these manifolds. Cohomology groups associated with complexes of differential forms on Mn in the presence of such a closed multi-symplectic form ϕl and their relations with the de Rham cohomologies of M are investigated. We show rigidity of a class of strongly associative (resp. strongly coassociative) submanifolds. We include an appendix describing all connected simply connected complete Riemannian manifolds admitting a parallel 3-form.

Nonstandard finite difference variational integrators for nonlinear Schrödinger equation with variable coefficients

In this paper, the idea of nonstandard finite difference discretization is employed to develop two variational integrators for the nonlinear Schrödinger equation with variable coefficients. These integrators are naturally multi-symplectic, and their multi-symplectic structures are presented by the multi-symplectic form formulas. Local truncation errors and convergences of the integrators are briefly discussed. The effectiveness and efficiency of the proposed schemes, such as the convergence order, numerical stability, and the capability in preserving the norm conservation, are verified in the numerical experiments.

Multi-symplectic wavelet splitting method for the strongly coupled Schrödinger system

We propose a multi-symplectic wavelet splitting method to solve the strongly coupled nonlinear Schrödinger equations. Based on its multi-symplectic formulation, the strongly coupled nonlinear Schrödinger equations can be split into one linear multi-symplectic subsystem and one nonlinear infinite-dimensional Hamiltonian subsystem. For the linear subsystem, the multi-symplectic wavelet collocation method and the symplectic Euler method are employed in spatial and temporal discretization, respectively. For the nonlinear subsystem, the mid-point symplectic scheme is used. Numerical simulations show the effectiveness of the proposed method during long-time numerical calculation.

Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs

Nonlinear wave equations, such as Burgers equation and compound KdV–Burgers equation, are a class of partial differential equations (PDEs) with dissipation in Hamiltonian space, the numerical method of which plays an important role in complex fluid analysis. Based on the multi-symplectic idea, a new theoretical framework named generalized multi-symplectic integrator for a class of nonlinear wave PDEs with small damping is proposed in this paper. The generalized multi-symplectic formulation is introduced, and a twelve-point generalized multi-symplectic scheme, which satisfies two discrete modified conservation laws approximately as well as the local momentum conservation law accurately, is constructed to solve the first-order PDEs that derived from the compound KdV–Burgers equation. To test the generalized multi-symplectic scheme, several numerical experiments on the travelling front solution are carried out, the results of which imply that the generalized multi-symplectic scheme can simulate the travelling front solution accurately and satisfy the modified conservation laws well when step sizes and the damping parameter satisfy the inequality (41). It is more remarkable that the scheme (36) can be used to capture the shock wave structure of the compound KdV–Burgers equation within one data point, which can further illustrate the good structure-preserving property of the generalized multi-symplectic scheme (36). From the results of this paper, we can conclude that, similar to the multi-symplectic scheme, the generalized multi-symplectic scheme also has two remarkable advantages: the excellent long-time numerical behavior and the good conservation property. For the existing of the excellent numerical properties, the generalized multi-symplectic method can be used to exposit some specific phenomena in the complex fluid.

Explicit Multisymplectic Fourier Pseudospectral Scheme for the Klein—Gordon—Zakharov Equations

Applying the Fourier pseudospectral method to space derivatives and the symplectic Euler rule to time derivatives in the multisymplectic form of the Klein-Gordon-Zakharov equations, we derive an explicit multisymplectic scheme. The semi-discrete energy and momentum conservation laws are given. Some numerical experiments are carried out to show the accuracy of the numerical solutions. The performance of the scheme in preserving the global energy and momentum conservation laws are also checked.

The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories

We introduce a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Second, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit Euler-Lagrange equations for fields obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Finally, we show a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields, Maxwell's equations, and elastostatics.

Numerical integration of the Ostrovsky equation based on its geometric structures

We consider structure preserving numerical schemes for the Ostrovsky equation, which describes gravity waves under the influence of Coriolis force. This equation has two associated invariants: an energy function and the L2 norm. It is widely accepted that structure preserving methods such as invariants-preserving and multi-symplectic integrators generally yield qualitatively better numerical results. In this paper we propose five geometric integrators for this equation: energy-preserving and norm-preserving finite difference and Galerkin schemes, and a multi-symplectic integrator based on a newly found multi-symplectic formulation. A numerical comparison of these schemes is provided, which indicates that the energy-preserving finite difference schemes are more advantageous than the other schemes.

Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear Klein‐Gordon equation with a triangle discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.

Multi-Symplectic Wavelet Collocation Method for Maxwell’s Equations

AbstractIn this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell’s equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.

Analyzing dynamic response of non-homogeneous string fixed at both ends

To survey the local conservation properties of complex dynamic problems, a structure-preserving numerical method, named as generalized multi-symplectic method, is proposed to analyze the dynamic response of a non-homogeneous string fixed at both ends. Firstly, based on the multi-symplectic idea, a generalized multi-symplectic form derived from the vibration equation of a non-homogeneous string is presented. Secondly, several conservation laws are deduced from the generalized multi-symplectic form to illustrate the local properties of the system. Thirdly, a centered box difference scheme satisfying the discrete local momentum conservation law exactly, named as a generalized multi-symplectic scheme, is constructed to analyze the dynamic response of the non-homogeneous string fixed at both ends. Finally, numerical experiments on the generalized multi-symplectic scheme are reported. The results illustrate the high accuracy, the good local conservation properties as well as the excellent long-time numerical behavior of the generalized multi-symplectic scheme well.

Multi-symplectic formulation of near-local Hamiltonian balanced models

We transform near-local Hamiltonian balanced models (HBMs) describing nearly geostrophic fluid motion (with constant Coriolis parameter) into multi-symplectic (MS) systems. This allows us to determine conservation of Lagrangian momentum, energy and potential vorticity for Salmon's L 1 dynamics; a similar approach works for other near-local balanced models (such as the -model). The MS approach also enables us to determine a class of systems that have a contact structure similar to that of the semigeostrophic model. The contact structure yields a contact transformation that makes the problem of front formation tractable. The new class includes the first local model with a variable Coriolis parameter that preserves all of the most useful geometric features of the semigeostrophic model.