Numerical Study Of Energy Conservation Methods In Several Partial Differential Equations

In mathematics and physics,there is a large class of partial differential equations,such as the Ito type coupled KdV equation,the nonlinear four order Schrodinger equation and the three coupling equation set.These equations have the conservation of energy.In recent years,the numerical method of the conservation of energy conservation of the differential equations has attracted much attention in the field of preserving structural algorithms.Compared with the mean vector field method,the two order mean vector field method can maintain the inherent energy conservation characteristics of the differential equation.Based on the thought of the modified vector field,G.R.W.Quispel et al.Put forward the high order mean vector field method,and this method is seldom studied in China.Therefore,we use this method to study the energy conservation properties of differential equations.It is progressiveness.In the method of preserving structure,the discrete line integration method has also begun to pay attention.Especially when the system is the SARS Hamilton system,the results obtained by the ordinary method are not ideal for the energy conservation.Using the discrete line integral method,the energy conservation scheme of the system can not only be directly constructed,but also the energy conservation of the equation is very accurate.This paper mainly studies the numerical applications of high-order mean vector field method and discrete line integral method in energy conservation partial differential equations.In the first chapter,we use the four order mean vector field method and the pseudospectral method to obtain the high order guaranteed energy scheme for the Ito type coupling KdV equation,and use this scheme to simulate the evolution behavior of the solitary wave under different parameters.The final simulation results show that the new high order energy conservation scheme can well simulate the behavior of the solitary wave of the Ito coupled KdV equation and keep the energy conservation characteristics of the equation accurately.In the second chapter,by constructing the high order energy preserving scheme of the nonlinear four order Schrodinger equation,and using the constructed high order energy conservation scheme to simulate the evolution behavior of the solitary wave of the equation,the simulation results show that the new scheme has good stability,and has a great advantage for the evolution behavior of the simulated solitary wave.At the same time,the discrete energy conservation characteristic of the equation is maintained.In the third chapter,because of the energy conservation of the three coupled Schrodinger equations,the three coupled Schrodinger equation group is first written as a classical Hamilton system,and then a new format of the three coupled Schrodinger equation is obtained by using the Boole discrete line integration method.Finally,a new format is used to simulate the behavior of the solitary wave under different parameters.The numerical results are shown to show the numerical results.It is shown that the energy conservation of the equation can be maintained by the discrete line integral method.