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 di...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.展开更多
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 solito...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.展开更多
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 app...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.展开更多
The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux...The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux in time into the integration in space. Compared with the traditional semi-Lagrange scheme, the scheme devised here tries to directly evaluate the average fluxes along cell edges. It is this difference that makes the scheme in this paper simple to implement and easily extend to nonlinear cases. The procedure of evaluation of the average fluxes only depends on the high-order spatial interpolation. Hence the scheme can be implemented as long as the spatial interpolation is available, and no additional temporal discretization is needed. In this paper, the high-order spatial discretization is chosen to be the classical 5th-order weighted essentially non-oscillatory spatial interpolation. In the end, 1D and 2D numerical results show that this method is rather robust. In addition, to exhibit the numerical resolution and efficiency of the proposed scheme, the numerical solutions of the classical 5th-order WENO scheme combined with the 3rd-order Runge-Kutta temporal discretization (WENOJS) are chosen as the reference. We find that the scheme proposed in the paper generates comparable solutions with that of WENOJS, but with less CPU time.展开更多
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 variabl...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.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.91130013)the Open Foundation of State Key Laboratory of HighPerformance Computing of China
文摘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.
基金Project supported by the National Natural Science Foundation of China(Grant No.11161017)the National Science Foundation of Hainan Province,China(Grant No.113001)
文摘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.
基金the National Key R&D Program of China(Grant No.2020YFA0709800)the National Natural Science Foundation of China(Grant Nos.11901577,11971481,12071481,and 12001539)+3 种基金the Natural Science Foundation of Hunan,China(Grant Nos.S2017JJQNJJ0764 and 2020JJ5652)the fund from Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering(Grant No.2018MMAEZD004)the Basic Research Foundation of National Numerical Wind Tunnel Project,China(Grant No.NNW2018-ZT4A08)the Research Fund of National University of Defense Technology(Grant No.ZK19-37)。
文摘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.
文摘The paper is devised to propose finite volume semi-Lagrange scheme for approximating linear and nonlinear hyperbolic conservation laws. Based on the idea of semi-Lagrangian scheme, we transform the integration of flux in time into the integration in space. Compared with the traditional semi-Lagrange scheme, the scheme devised here tries to directly evaluate the average fluxes along cell edges. It is this difference that makes the scheme in this paper simple to implement and easily extend to nonlinear cases. The procedure of evaluation of the average fluxes only depends on the high-order spatial interpolation. Hence the scheme can be implemented as long as the spatial interpolation is available, and no additional temporal discretization is needed. In this paper, the high-order spatial discretization is chosen to be the classical 5th-order weighted essentially non-oscillatory spatial interpolation. In the end, 1D and 2D numerical results show that this method is rather robust. In addition, to exhibit the numerical resolution and efficiency of the proposed scheme, the numerical solutions of the classical 5th-order WENO scheme combined with the 3rd-order Runge-Kutta temporal discretization (WENOJS) are chosen as the reference. We find that the scheme proposed in the paper generates comparable solutions with that of WENOJS, but with less CPU time.
基金supported by the Innovative Science Research Project for Grad-uate Students of Hainan Province(Grant Nos.Hys2014-17)the Visiting Project of Hainan University and the Fostering Program of Excellent Dissertation for the Gradu-ate Students of Hainan University,the Natural Science Foundation of China(Grant Nos.11161017,11561018)+1 种基金the National Science Foundation of Hainan Province(Grant Nos.114003)the Training Programs of Innovation and Entrepreneurship for Under-graduates of Hainan University.
文摘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.
