Structure-preserving Property Research Articles

Reduction and Integrability of Stochastic Dynamical Systems

This paper is devoted to the study of qualitative geometrical properties of stochastic dynamical systems, namely their symmetries, reduction, and integrability. In particular, we show that an SDS that is diffusion-wise symmetric with respect to a proper Lie group action can be diffusion-wise reduced to an SDS on the quotient space. We also show necessary and sufficient conditions for an SDS to be projectable via a surjective map. We then introduce the notion of integrability of SDS’s, and extend the results on the existence and structure-preserving property of Liouville torus actions from the classical case to the case of integrable SDS’s. We also show how integrable SDS’s are related to compatible families of integrable Riemannian metrics on manifolds.

Structure-preserving properties of Störmer-Verlet scheme for mathematical pendulum

The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conservation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Stormer-Verlet scheme is first constructed in a Hamiltonian framework. The conservation law of the Stormer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Stormer-Verlet scheme associated with the conservation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the Stormer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the Stormer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Stormer-Verlet scheme.

A symplectic method for structure-preserving modelling of damped acoustic waves

In this paper, a symplectic method for structure-preserving modelling of the damped acoustic wave equation is introduced. The equation is traditionally solved using non-symplectic schemes. However, these schemes corrupt some intrinsic properties of the equation such as the conservation of both precision and the damping property in long-term calculations. In the method presented, an explicit second-order symplectic scheme is used for the time discretization, whereas physical space is discretized by the discrete singular convolution differentiator. The performance of the proposed scheme has been tested and verified using numerical simulations of the attenuating scalar seismic-wave equation. Scalar seismic wave-field modelling experiments on a heterogeneous medium with both damping and high-parameter contrasts demonstrate the superior performance of the approach presented for suppression of numerical dispersion. Long-term computational experiments display the remarkable capability of the approach presented for long-time simulations of damped acoustic wave equations. Promising numerical results suggest that the approach is suitable for high-precision and long-time numerical simulations of wave equations with damping terms, as it has a structure-preserving property for the damping term.

Variational discretization of constrained Birkhoffian systems

In this paper, we derive a variational characterization of constrained Birkhoffian dynamics in both continuous and discrete settings. When additional algebraic constraints appear, derivation of the necessary conditions under which the Pfaff action is extremized gives constrained Birkhoffian equations. Inspired by this continuous framework, we directly discretize the constraints as well as the Pfaff action and consequently formulate the discrete constrained Birkhoffian dynamics. Via this discrete variational approach which is parallel with the continuous case, the resulting discrete constrained Birkhoffian equations automatically preserve the intrinsic symplectic structure when identified as numerical algorithms. Considering that the obtained algorithms require not only the specification of an initial configuration but also a second configuration to operate, we present a natural, reasonable, and efficient method of initialization of simulations. While retaining the structure-preserving property, the obtained discrete schemes exhibit excellent numerical behaviors, demonstrated by numerical examples dealing with the mathematical pendulum and the 3D pendulum.

Real Fast Structure-Preserving Algorithm for Eigenproblem of Complex Hermitian Matrices

It is well known that the flops for complex operations are usually 4 times of real cases. In the paper, using real operations instead of complex, a real fast structure-preserving algorithm for eigenproblem of complex Hermitian matrices is given. We make use of the real symmetric and skew-Hamiltonian structure transformed by Wilkinson's way, focus on symplectic orthogonal similarity transformations and their structure-preserving property, and then reduce it into a two-by-two block tridiagonal symmetric matrix. Finally a real algorithm can be quickly obtained for eigenvalue problems of the original Hermitian matrix. Numerical experiments show that the fast algorithm can solve real complex Hermitian matrix efficiently, stably, and with high precision.

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.

Structure-preserving modelling of elastic waves: a symplectic discrete singular convolution differentiator method

SUMMARY In this paper, we introduce the so-called symplectic discrete singular convolution differentiator (SDSCD) method for structure-preserving modelling of elastic waves. In the method presented, physical space is discretized by the DSCD, whereas an explicit third-order symplectic scheme is used for the time discretization. This approach uses optimization and truncation to form a localized operator. This preserves the fine structure of the wavefield in complex media and avoids non-causal interaction when parameter discontinuities are present in the medium. Theoretically, the approach presented is a structure-preserving algorithm. Also, some numerical experiments are shown in this paper. Elastic wavefield modelling experiments on a laterally heterogeneous medium with high parameter contrasts demonstrate the superior performance of the SDSCD for suppression of numerical dispersion. Long-term computational experiments exhibit the remarkable capability of the approach presented for long-time simulations. Promising numerical results suggest the SDSCD is suitable for high-precision and long-time numerical simulations, as it has structure-preserving property and it can suppress effectively numerical dispersion when coarse grids are used.

Numerical Method with the Third-Order ENO Reconstruction Satisfying Two Conversation Laws for Linear Advection Equation

In recent years, Mao and his co-workers developed a class of finite-volume schemes for evolution partial differential equations, see [1-5].The schemes show a super-convergence quality and have good structure-preserving property in long-time numerical simulations. In [6], Chen and Ma developed a scheme which combine the idea of paper [5] and that of the the second-order ENO scheme [7]. In this paper, we propose a scheme which extend the result of [6] and obtain the scheme using the third-order ENO reconstruction. Numerical experiments show that our scheme is robust in long-time behaviors. Numerical solutions are far better than those of [6].

Scalar Seismic-Wave Equation Modeling by a Multisymplectic Discrete Singular Convolution Differentiator Method

High-precision modeling of seismic-wave propagation in heterogeneous media is very important to seismological investigation. However, such modeling is one of the difficult problems in the seismological research fields. For developing methods of seismic inversion and high-resolution seismic-wave imaging, the modeling problem must be solved as perfectly as possible. Moreover, for long-term computations of seismic waves (e.g., Earth’s free-oscillations modeling and seismic noise-propagation modeling), the capability of seismic modeling methods for long-time simulations is in great demand. In this paper, an alternative method for accurately and efficiently modeling seismic wave fields is presented; it is based on a multisymplectic discrete singular convolution differentiator scheme (MDSCD). This approach uses optimization and truncation to form a localized operator, which preserves the fine structure of the wave field in complex media and avoids noncausal interaction when parameter discontinuities are present in the medium. The approach presented has a structure-preserving property, which is suitable for treating questions of high-precision or long-time numerical simulations. Our numerical results indicate that this method can suppress numerical dispersion and allow for research into long-time numerical simulations of wave fields. These numerical results also show that the MDSCD method can effectively capture the inner interface without any special treatment at the discontinuity.

Modeling of the Seismic Scalar Wave Field Using the Symplectic Scheme and Singular Kernel Convolutional Differentiator

AbstractThis work adopts both the symplectic difference scheme and singular kernel convolutional differentiator for modeling of the seismic scalar wave field. In this method the computational accuracy and stability have been greatly improved with a slight increase of calculation amount. Compared with other non‐symplectic numerical methods, major advantages of the method presented in this paper are the structure‐preserving property and a strong capability of long‐time tracing. This approach provides a new and effective choice for high‐accuracy modeling of large‐scale and long‐time history seismic wave fields.

Numerical Method of Symplectic State Transition Matrix and Application to Fully Perturbed Earth Orbit

This paper presents a numerical method for deriving a symplectic state transition matrix for high-fidelity Earth orbits subject to non-dissipative perturbation forces. By taking advantage of properties of Hamiltonian systems, this method provides an exact solution space mapping of linearized orbital dynamics, preserving the symplectic structure that all Hamiltonian systems should possess by nature. This method can be applied to accurate, yet computationally efficient dynamic filters, long-term propagations of the motions of formation flying spacecraft and the eigenstructure analysis of N-body dynamics, etc., when the exact structure-preserving property is crucial. We show the derivation of the numerical method of symplectic state transition matrix, and apply it to Earth orbits with perturbation forces based on real ephemerides. These numerical examples reveal that this method shows improvements in preserving the structural properties of the state transition matrix, and in the computational efficiency compared to the conventional linear state transition matrix with Euler or Runge-Kutta integration.

Computation and Applications of an Orbital Dynamics Symplectic State Transition Matrix

This paper presents a numerical method for deriving a symplectic state transition matrix for an arbitrary Hamiltonian dynamical system. It provides the exact solution-space mapping of the linearized Hamiltonian systems, preserving the symplectic structure that all Hamiltonian systems should possess by nature. The symplectic state transition matrix can be applied to accurate, yet computationally efficient, dynamic filters, long-term propagations of the motions of formation-flying spacecraft, eigenstructure/manifold analysis of N-body dynamics, etc., when the exact structure-preserving property is crucial. We present the derivation and key characteristics of the symplectic state transition matrix and apply it to the two-body dynamics, the circular restricted three-body problem, and an Earth orbit with perturbation forces based on the real ephemeris. These numerical examples reveal that this numerical symplectic state transition matrix shows improvements in preserving the structural properties of the state transition matrix as compared with the conventional linear state transition matrix with Euler or Runge―Kutta integrations.

A structure-preserving algorithm for the minimum H ∞ norm computation of finite-time state feedback control problem

The algorithm being developed here is based on the generating function approach for finite-time H ∞ control and application of canonical transformation of linear Hamiltonian system. First, an equivalent finite-time H ∞ control law in terms of the third-type generating function is derived. Then, by using symplectic structure of the Hamiltonian system's state transition matrix, a group of matrix recursive formulae are deduced for the evaluation of the finite-time H ∞ control law. Combining with a matrix singularity testing procedure, this recursive algorithm verifies the existence condition of sub-optimal H ∞ controllers and gives the minimum H ∞ norm of finite-time control systems. Inherited from the canonical transformation, the matrix recursive formulae have a standard symplectic form; this structure-preserving property helps facilitate reliable and effective computation. Numerical results show the effectiveness of the proposed algorithm.

An eigenelement method and two homogenization conditions

Under inspiration from the structure-preserving property of symplectic difference schemes for Hamiltonian systems, two homogenization conditions for a representative unit cell of the periodical composites are proposed, one condition is the equivalence of strain energy, and the other is the deformation similarity. Based on these two homogenization conditions, an eigenelement method is presented, which is characteristic of structure-preserving property. It follows from the frequency comparisons that the eigenelement method is more accurate than the stiffness average method and the compliance average method.

Multisymplectic Structures and Discretizations for One-way Wave Equations

Multisymplectic structures for one-way wave equations are presented in this letter. Based on the multisymplectic formulation, we obtain the corresponding multisymplectic discretizations. The structure-preserving property of a finite difference scheme for the first-order one-way wave equation is proved. Implications and applications of this result are explored.

Schur complements of Bezoutians and the inversion of block Hankel and block Toeplitz matrices

We derive a closed inversion formula for an np × np square block Hankel matrix H n − 1 = ( W i − j ) with entries W i from the ring of the p × p matrices over a field. The representation of H −1 n − 1 relies upon a strong structure-preserving property of the Schur complements of the nonsingular leading principal submatrices of a certain generalized Bezoutian of matrix polynomials. These polynomials may be determined by computing iteratively the Schur complements of the nonsingular leading principal submatrices of H n − 1 in an augmented block Hankel matrix H ̃ which is a Bezoutian matrix up to multiplication by proper reverse matrices. Since the special structure of the Bezoutian matrix is preserved at any step of the computation, such method can be efficiently implemented at an overall computational cost of O( p 3 n 2) operations in the average case and O( p 3 n 3) operations in the worst case. In the normal case, a suitable modification which uses both divide and conquer strategies and fast polynomial arithmetic reaches the bound of O( p 3 n log 2 n) arithmetical operations.