Average vector field methods for the coupled Schrdinger KdV equations

The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.

Novel energy dissipative method on the adaptive spatial discretization for the Allen–Cahn equation

We propose a novel energy dissipative method for the Allen–Cahn equation on nonuniform grids.For spatial discretization,the classical central difference method is utilized,while the average vector field method is applied for time discretization.Compared with the average vector field method on the uniform mesh,the proposed method can involve fewer grid points and achieve better numerical performance over long time simulation.This is due to the moving mesh method,which can concentrate the grid points more densely where the solution changes drastically.Numerical experiments are provided to illustrate the advantages of the proposed concrete adaptive energy dissipative scheme under large time and space steps over a long time.

A high order energy preserving scheme for the strongly coupled nonlinear Schr¨odinger system

A high order energy preserving scheme for a strongly coupled nonlinear Schrōdinger system is roposed by using the average vector field method. The high order energy preserving scheme is applied to simulate the soliton evolution of the strongly coupled Schrōdinger system. Numerical results show that the high order energy preserving scheme can well simulate the soliton evolution, moreover, it preserves the discrete energy of the strongly coupled nonlinear Schrōdinger system exactly.

High Order Energy-Preserving Method of the “Good” Boussinesq Equation

The fourth order average vector field(AVF)method is applied to solve the“Good”Boussinesq equation.The semi-discrete system of the“good”Boussi-nesq equation obtained by the pseudo-spectral method in spatial variable,which is a classical finite dimensional Hamiltonian system,is discretizated by the fourth order average vector field method.Thus,a new high order energy conservation scheme of the“good”Boussinesq equation is obtained.Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the“good”Boussinesq equation exactly and simulate evolution of different solitary waves well.

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preservmg formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq＂ （t）＋ Bq（t） = f（q（t））, where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f（q） = -VqV（q） for a real-valued function V（q）. The energy-preserving formulae can exactly preserve the Hamiltonian H（q＇,q） = 1/2q＇^TAq＇＋ 1/2q^TBq - V（q）. We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.

An Energy-Preserving Scheme for the Coupled Gross-Pitaevskii Equations

An energy-preserving scheme is proposed for the coupled Gross-Pitaevskii equations.The scheme is constructed by high order compact method in the spatial direction and average vector field method in the temporal direction,respectively.The scheme is energy-preserving,stable,and of sixth order in space and of second order in time.Numerical experiments verify the theoretical results.The dynamic behavior modeled by the coupled Gross-Pitaevskii equations is also numerically investigated.