High Order Conserving Numerical Methods For Some Classes Of Nonlinear Partial Differential Equations

The mathematical models of many scientific and engineering problems are constructed by partial differential equations.However,most of the partial differential equations have no ana-lytical solution,which brings great difficulties to solve the practical engineering transformation and engineering control design problems.Therefore,so it is particularly important to construct numerical solutions for partial differential equations.In addition,in scientific and engineering calculations,it is often required that the numerical solution has high accuracy,keep some prop-erties of the original model,such as energy conservation,and that the numerical error will not be too large under long time simulation.The high order conservation numerical scheme can meet these“harsh”requirements.In this paper,the Klein-Gordon equation and the Korteweg-de Vries Benjamin-Bona-Mahony(KdV-BBM)equation are solved numerically by the variable limit in-tegral method,and the Benjamin-Bona-Mahony(BBM)equation and the improved Boussinesq equation are solved numerically by the local discontinuous Galerkin method,and the corre-sponding numerical schemes with high accuracy are obtained to preserve the conservation of the original equation.These equations appear in important scientific and engineering fields,such as fluid mechanics,nonlinear optics,acoustics,quantum physics and so on.Therefore,the high order conservation numerical schemes of these equations will not only help the theoretical development of related fields,but also have a wide range of applications.The main innovative results of this paper are as follows:1.The theory of the integral method with variational limit is developed and perfected in the following two part.First,we discuss how to deal with the variable limit integral by using the Taylor formula method,and how to design the integral limit parameters to obtain the”neat”vari-able limit integral results.Through Taylor formula method,it is shown that all the schemes that can be obtained from the finite difference also can be obtained by the integral method with vari-ational limit.Secondly,through the operation of the exchange integration order of the variable limit integral,it is revealed that the integral method with variational limit uses the”weighted”information of all the points near the grid points,not just the information on the grid points.This is essentially different from the information that the difference method only uses on the grid nodes.2.By using the integral method with variational limit,a fourth-order compact conservation spatial semi-discrete scheme for nonlinear Klein-Gordon equation is designed,and the stability and convergence of the spatial semi-discrete scheme are proved.Then the fully discrete scheme is obtained by using the multi-dimensional extended Runge-Kutta-Nystr”o m(ERKN)method to discretize the time.In addition,through a numerical example,the convergence order and energy conservation are verified.Compared with some other methods,it is found that the in-tegral method with variational limit has smaller error and smaller energy difference.The most important contributions of numerical example part is the finite time blow-up of the solution is simulated,and the influence of initial energy and functional on the blow-up time is analyzed.The proposed fourth-order conservation numerical scheme will not only have important theo-retical and application value,but also provide an important reference for practical engineering control problems such as how to avoid blasting or controlled blasting(advance or delay blasting time).3.By using the integral method with variational limit,two fourth-order spatial semi-discrete schemes for nonlinear KdV-BBM equations are obtained,which keep the conservation of mass and energy.It is proved that the solutions of these two spatial semi-discrete schemes are stable according to the initial value under the discrete infinite norm and converge to the exact so-lution according to O(h~4).Then the implicit midpoint method is used to discretize the time,and the fully discrete scheme is obtained.Numerical experiments verify the order of convergence in time and space,the conservation of mass and energy of the fully discrete scheme,and find that the error of the scheme increases slowly over a long period of time.Finally,the collision of two soliton waves is simulated.4.By using the local discontinuous Galerkin method,two kinds of numerical schemes with optimal prior error estimates,LDG scheme and d LDG scheme,are proposed,analyzed and numerically verified for the BBM equation.The LDG scheme can keep the mass conservation of the discrete form.By selecting the appropriate numerical flux,the LDG scheme can also preserve/dissipate the energy in the discrete form.The d LDG scheme is constructed by the idea of“doubling PDE”,that is,introducing a zero-solution BBM equation.The d LDG scheme can also preserve discrete mass and energy.An important work of this paper is to reveal the relationship between the errors of auxiliary variables and principal variables.Using this relation,the nonlinear term is constrained by auxiliary variables,and it is proved that the two kinds of numerical schemes have optimal a priori error estimates.The implicit midpoint method which can preserve energy is used in time discretization.Numerical experiments show that the energy conservation LDG scheme is better than the energy attenuation LDG method in terms of long-time error,waveform preservation and phase error.On the one hand,the d LDG method can improve the results of the suboptimal error estimation of the Central-LDG scheme,on the other hand,compared with the conservative LDG scheme,the numerical error of the d LDG method is smaller under the same grid conditions,but the computing time is not much different.5.The local discontinuous Galerkin method is used to discretize the improved Boussinesq equation,and a LDG scheme is proposed,which can preserve the mass and energy of the orig-inal equation and has the optimal error estimate.Then two fully discrete schemes which can accurately preserve mass and energy are obtained by using explicit and implicit time discretiza-tion methods.Numerical examples show that the method has the optimal convergence order.The numerical simulation of wave propagation shows that the proposed LDG scheme can well simulate the propagation of single wave,the collision of two waves,the splitting of single wave and finite time blow-up.