Global Energy Conservation Laws Research Articles

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.

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.

A local energy-preserving scheme for Zakharov system**Project supported by the National Natural Science Foundation of China (Grant No. 11771213) and the Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology (Grant No. 2243141701090).

In this paper, we propose a local conservation law for the Zakharov system. The property is held in any local time-space region which is independent of the boundary condition and more essential than the global energy conservation law. Based on the rule that the numerical methods should preserve the intrinsic properties as much as possible, we propose a local energy-preserving (LEP) scheme for the system. The merit of the proposed scheme is that the local energy conservation law can be conserved exactly in any time-space region. With homogeneous Dirchlet boundary conditions, the proposed LEP scheme also possesses the discrete global mass and energy conservation laws. The theoretical properties are verified by numerical results.

A new local energy-preserving algorithm for the BBM equation

In this paper, a new local energy-preserving algorithm is proposed based on the temporal and spatial discretizations where the Average Vector Field (AVF) method and the implicit midpoint method are used for the temporal discretization and spatial discretization, respectively. In any local time-space region, the local mass and local energy are conserved by the algorithm. With periodic boundary conditions, it is worth noting that the global mass and global energy conservation law are also admitted. Numerical experiments are performed to support our theoretical analysis and show the conservation properties of the algorithm intuitively.

Conservative modified Crank–Nicolson and time-splitting wavelet methods for modeling Bose–Einstein condensates in delta potentials

This paper explores two wavelet-based energy-conserving algorithms for the Gross–Pitaevskii equation with delta potentials in Bose–Einstein condensates, named modified Crank–Nicolson wavelet method and time-splitting wavelet method, respectively. Both proposed methods can preserve the intrinsic properties of original problems as much as possible. Meanwhile, the rigorous error estimates and some conservative properties are investigated. They are proved to preserve the charge conservation exactly. The global energy conservation laws can be preserved under several conditions. In practical computations, to avoid a large drift in energy values caused by discontinuous potential well, an improved discrete delta function is implemented. Numerical experiments for attractive and repulsive cases are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.

A local energy-preserving scheme for Klein–Gordon–Schrödinger equations**Project supported by the National Natural Science Foundation of China (Grant Nos. 11201169, 11271195, and 41231173) and the Project of Graduate Education Innovation of Jiangsu Province, China (Grant No. CXLX13 366).

A local energy conservation law is proposed for the Klein–Gordon–Schrödinger equations, which is held in any local time–space region. The local property is independent of the boundary condition and more essential than the global energy conservation law. To develop a numerical method preserving the intrinsic properties as much as possible, we propose a local energy-preserving (LEP) scheme for the equations. The merit of the proposed scheme is that the local energy conservation law can hold exactly in any time–space region. With the periodic boundary conditions, the scheme also possesses the discrete change and global energy conservation laws. A nonlinear analysis shows that the LEP scheme converges to the exact solutions with order \U0001d4aa(τ2 + h2). The theoretical properties are verified by numerical experiments.

LOD-MS for Gross-Pitaevskii Equation in Bose-Einstein Condensates

AbstractThe local one-dimensional multisymplectic scheme (LOD-MS) is developed for the three-dimensional (3D) Gross-Pitaevskii (GP) equation in Bose-Einstein condensates. The idea is originated from the advantages of multisymplectic integrators and from the cheap computational cost of the local one-dimensional (LOD) method. The 3D GP equation is split into three linear LOD Schrödinger equations and an exactly solvable nonlinear Hamiltonian ODE. The three linear LOD Schrödinger equations are multisymplectic which can be approximated by multisymplectic integrator (MI). The conservative properties of the proposed scheme are investigated. It is mass-preserving. Surprisingly, the scheme preserves the discrete local energy conservation laws and global energy conservation law if the wave function is variable separable. This is impossible for conventional MIs in nonlinear Hamiltonian context. The numerical results show that the LOD-MS can simulate the original problems very well. They are consistent with the numerical analysis.

Numerical dispersion analysis of a multi-symplectic scheme for the three dimensional Maxwell’s equations

In this paper, we study a multi-symplectic scheme for three dimensional Maxwell’s equations in a simple medium. This is a system of PDEs with multi-symplectic structures. We prove that this multi-symplectic scheme preserves the discrete version of local and global energy conservation law and the discrete divergence. Furthermore, we extend the discussion to several dispersion properties of the multi-symplectic scheme including the numerical dispersion relation, the numerical group velocity, the effect of large time steps and the CFL condition.

Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations

The exact global energy conservation laws for the full and truncated versions of the nonlinear electromagnetic gyrokinetic equations in general magnetic geometry are presented. In each version, the relation between polarization and magnetization effects in the gyrokinetic Poisson and Ampère equations and the quadratic ponderomotive gyrocenter Hamiltonian is emphasized.

A note on multisymplectic Fourier pseudospectral discretization for the nonlinear Schrödinger equation

Based on the multisymplectic Fourier pseudospectral scheme for the nonlinear Schrödinger (NLS) equation, we investigate some discrete properties corresponding to local conservation laws of the original equation. The discrete normal conservation law is proved, and the error estimation of local and global energy conservation laws are also obtained. Numerical experiments for cubic NLS equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results.

On point blast theory

The problem of a spherical shock front generated in a point explosion and traveling in a homogeneous medium is analytically studied with account for counterpressure on the entire infinite interval of its existence. For this purpose, asymptotic representations of the excess pressure in the shock wave near and far away from the energy release point are matched. It is possible analytically to continue the four-term expansion for the far zone involving unknown constants, so that it rigorously coincides with the four-term power expansion of the solution for the singular point, that is, the blast center. The problem of determination of the unknown constants is mathematically closed by the derived entropy loss integral which expresses the global energy conservation law. The analytical dependence of the excess pressure in the shock wave on distance thus obtained is in good agreement with the results of numerical calculations.

Distributional curvature of time dependent cosmic strings

Colombeau's theory of generalized functions is used to calculate the contributions, at the rotation axis, to the distributional curvature for a time-dependent radiating cosmic string, and hence the mass per unit length of the string source. This mass per unit length is compared with the mass at null infinity, giving evidence for a global energy conservation law.

Reduced energy conservation law for magnetized plasma

A global energy conservation law for a magnetized plasma is studied within the context of a quasiparticle description. A reduced energy conservation law is derived for low-frequency, as compared to the gyromagnetic frequency, plasma motions with regard to both non-uniform mean flows and fluctuations in the plasma. The mean value of plasma energy is calculated and sufficient stability conditions for non-equilibrium plasmas are derived.