Multi-symplectic Conservation Law Research Articles

Functional Equivariance and Conservation Laws in Numerical Integration

Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs. We introduce the concept of functional equivariance, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.

Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique and the partitioned Runge–Kutta method. Specific numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schrödinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.

A charge-preserving compact splitting method for solving the coupled stochastic nonlinear Schrödinger equations

In this paper, a splitting method for solving a system of coupled stochastic nonlinear Schrödinger (CSNLS) equations is proposed. Based on its Hamiltonian structure, the CSNLS equations split into two subproblems such that one of them is linear. The solution of the nonlinear subproblem is computed exactly, and the linear subproblem is discritized through a compact method in the spatial direction and the symplectic implicit midpoint rule with respect to the time variable. Stability, discrete stochastic multi-symplectic conservation law, discrete charge and energy conservation laws for the proposed method are investigated. Numerical results are reported for various amplitudes of noise. We present the propagation and collision of solitons which are different from the deterministic case. Finally, the mean-square order of convergence of the proposed algorithm in the temporal and spatial directions for the CSNLS equations is investigated in a numerical example.

Three Kinds of Novel Multi-Symplectic Methods for Stochastic Hamiltonian Partial Differential Equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique, and the partitioned Runge-Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schr\"odinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.

The exponential invariant energy quadratization approach for general multi-symplectic Hamiltonian PDEs

Many conservative systems in physical sciences can be described by multi-symplectic Hamiltonian PDEs (MS-HPDEs) admitting three important inherent properties named as multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we develop a novel strategy to systematically derive linearly implicit local energy-preserving schemes for general MS-HPDEs, which is named the exponential invariant energy quadratization approach (EIEQ). Such novel strategy is based on the exponential form of non-quadratic terms of the state function that can remove the bounded-from-below restriction, so it is more applicable than the traditional IEQ approach widely employed to construct linearly implicit schemes. Moreover, we provide a completely explicit discretization of the auxiliary variable combined with the nonlinear term, which obtains linearly implicit schemes of the constant coefficient, making their more effective than the IEQ schemes for rapid simulations. We also present the multiple EIEQ approach to improve the applicability of the EIEQ approach for MS-HPDEs with more non-quadratic terms. In addition, when periodic or homogeneous boundary conditions are considered, the proposed schemes are global energy-preserving and can be explicitly solved by using the fast Fourier transform. Finally, ample numerical tests are carried out to demonstrate the computational efficiency, conservation and accuracy of EIEQ schemes.

Multisymplectic Hamiltonian Variational Integrators

Variational integrators have traditionally been constructed from the perspective of Lagrangian mechanics, but there have been recent efforts to adopt discrete variational approaches to the symplectic discretization of Hamiltonian mechanics using Hamiltonian variational integrators. In this paper, we will extend these results to the setting of Hamiltonian multisymplectic field theories. We demonstrate that one can use the notion of Type II generating functionals for Hamiltonian partial differential equations as the basis for systematically constructing Galerkin Hamiltonian variational integrators that automatically satisfy a discrete multisymplectic conservation law, and establish a discrete Noether's theorem for discretizations that are invariant under a Lie group action on the discrete dual jet bundle. In addition, we demonstrate that for spacetime tensor product discretizations, one can recover the multisymplectic integrators of Bridges and Reich, and show that a variational multisymplectic discretization of a Hamiltonian multisymplectic field theory using spacetime tensor product Runge–Kutta discretizations is well-defined if and only if the partitioned Runge-Kutta methods are symplectic in space and time.

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.

Stochastic conformal schemes for damped stochastic Klein-Gordon equation with additive noise

In this article, stochastic conformal schemes of the damped stochastic Klein-Gordon equation with additive noise are studied. It is shown that this equation possesses the stochastic conformal multi-symplectic conservation law. Under appropriate boundary conditions, the global momentum evolution law and the global energy evolution law are proposed. We chiefly develop the stochastic conformal Preissman scheme, the stochastic conformal discrete gradient scheme and the stochastic conformal Euler box scheme to preserve the geometric structures of the original system. Specifically, we make theoretical discussions on the three proposed schemes to obtain corresponding discrete conservation laws or discrete evolution laws. Then the damped stochastic linear Klein-Gordon equation and the damped stochastic nonlinear Klein-Gordon equation with cubic nonlinearity are taken as examples to demonstrate the validity of the proposed schemes. Through numerical experiments and comparisons, the superiorities of the proposed schemes are fully shown, which are consistent with our theoretical analysis. Moreover, the mean square convergence orders of the three stochastic conformal schemes in time direction and space direction are tested numerically.

Error estimates of structure‐preserving Fourier pseudospectral methods for the fractional Schrödinger equation

AbstractThis paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L2 norm with the second‐order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results.

Multi-symplectic quasi-interpolation method for Hamiltonian partial differential equations

In this paper, we propose a multi-symplectic quasi-interpolation method for solving multi-symplectic Hamiltonian partial differential equations. Based on the method of lines, we first discretize the multi-symplectic PDEs using quasi-interpolation method and then employ appropriate time integrators to obtain the full-discrete system. The local conservation properties including multi-symplectic conservation laws, energy conservation laws and momentum conservation laws are discussed in detail. For illustration, we provide two concrete examples: the nonlinear wave equation and the nonlinear Schrödinger equation. The salient feature of our multi-symplectic quasi-interpolation method is that it is valid both on uniform grids and nonuniform grids. The numerical results show the good accuracy and excellent conservation properties of the proposed method.

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.

Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin (HDG) method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the "hybridized" versions of several of the most commonly-used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structure-preserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily-high-order methods on unstructured meshes.

Stochastic multi-symplectic Runge–Kutta methods for stochastic Hamiltonian PDEs

In this paper, we consider stochastic Runge–Kutta methods for stochastic Hamiltonian partial differential equations and present some sufficient conditions for stochastic multi-symplecticity of stochastic Runge–Kutta methods. To present more clearly, we apply these ideas to three dimensional stochastic Maxwell equations driven by multiplicative noise, which play an important role in stochastic electromagnetism and statistical radiophysics areas. Theoretical analysis shows that the methods inherit the energy conservation law of the original system, and preserve the discrete stochastic multi-symplectic conservation law almost surely.

Structure-preserving algorithm for thermal conduction of local thermal nonequilibrium saturated porous medium

Based on the two-energy equation model and constitutive equations of local thermal nonequilibrium saturated porous media, one-dimensional thermal conduction equations of solid skeleton and pore fluid are established. Introducing orthogonal variables, a first order generalized multi-symplectic structure-preserving form of thermal conduction equations is derived as well as the errors of generalized multi-symplectic conservation law and local momentum. The temperature profiles of two phases are obtained and the effect of the heat exchange coefficient between two phases on the process from local thermal nonequilibrium to local thermal equilibrium is revealed numerically. According to the relative error between numerical solution and analytical solution derived from the separating variables method, it can be concluded that this structure-preserving scheme is a valid scheme with high accuracy. Numerical errors of generalized multi-symplectic conservation law and local momentum are presented for two typical types of the thermal conduction in one-dimensional saturated porous media. According to these numerical results, it can be concluded that this structure-preserving algorithm has long-time numerical stability and good conservation properties.

Structure-preserving numerical methods for the fractional Schrödinger equation

This paper considers the long-time integration of the nonlinear fractional Schrödinger equation involving the fractional Laplacian from the point of view of symplectic geometry. By virtue of a variational principle with the fractional Laplacian, the equation is first reformulated as a Hamiltonian system with a symplectic structure. Then, by introducing a pair of intermediate variables with a fractional operator, the equation is reformulated in another form for which more conservation laws are found. When reducing to the case of integer order, they correspond to multi-symplectic conservation law and local energy conservation law for the classic Schrödinger equation. After that, structure-preserving algorithms with the Fourier pseudospectral approximation to the spatial fractional operator are constructed. It is proved that the semi-discrete and fully discrete systems satisfy the corresponding symplectic or other conservation laws in the discrete sense. Numerical tests are performed to validate the efficiency of the methods by showing their remarkable conservation properties in the long-time simulation.

Novel Conformal Structure-Preserving Algorithms for Coupled Damped Nonlinear Schrödinger System

AbstractThis paper introduces two novel conformal structure-preserving algorithms for solving the coupled damped nonlinear Schrödinger (CDNLS) system, which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws. The proposed algorithms can preserve corresponding conformal multi-symplectic conservation law and conformal momentum conservation law in any local time-space region, respectively. Moreover, it is further shown that the algorithms admit the conformal charge conservation law, and exactly preserve the dissipation rate of charge under appropriate boundary conditions. Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.

An energy-conserving method for stochastic Maxwell equations with multiplicative noise

In this paper, it is shown that three-dimensional stochastic Maxwell equations with multiplicative noise are stochastic Hamiltonian partial differential equations possessing a geometric structure (i.e. stochastic multi-symplectic conservation law), and the energy of system is a conservative quantity almost surely. We propose a stochastic multi-symplectic energy-conserving method for the equations by using the wavelet collocation method in space and stochastic symplectic method in time. Numerical experiments are performed to verify the excellent abilities of the proposed method in providing accurate solution and preserving energy. The mean square convergence result of the method in temporal direction is tested numerically, and numerical comparisons with finite difference method are also investigated.

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

ABSTRACTThis 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.

A Compact Scheme for Coupled Stochastic Nonlinear Schrödinger Equations

AbstractIn this paper, we propose a compact scheme to numerically study the coupled stochastic nonlinear Schrödinger equations. We prove that the compact scheme preserves the discrete stochastic multi-symplectic conservation law, discrete charge conservation law and discrete energy evolution law almost surely. Numerical experiments confirm well the theoretical analysis results. Furthermore, we present a detailed numerical investigation of the optical phenomena based on the compact scheme. By numerical experiments for various amplitudes of noise, we find that the noise accelerates the oscillation of the soliton and leads to the decay of the solution amplitudes with respect to time. In particular, if the noise is relatively strong, the soliton will be totally destroyed. Meanwhile, we observe that the phase shift is sensibly modified by the noise. Moreover, the numerical results present inelastic interaction which is different from the deterministic case.

Multisymplectic method for the Camassa-Holm equation

The Camassa-Holm equation, a completely integrable evolution equation, contains rich geometric structures. For the existence of the bi-Hamiltonian structure and the so-called peaked wave solutions, considerable interest has been aroused in the last several decades. Focusing on local geometric properties of the peaked wave solutions for the Camassa-Holm equation, we propose the multisymplectic method to simulate the propagation of the peaked wave in this paper. Based on the multisymplectic theory, we present a multisymplectic formulation of the Camassa-Holm equation and the multisymplectic conservation law. Then, we apply the Euler box scheme to construct the structure-preserving scheme of the multisymplectic form. Numerical results show the merits of the multisymplectic scheme constructed, especially the local conservative properties on the wave form in the propagation process.