Study On Format Property And Application Of Integral Method With Variational Limit

Partial differential equations play an important role in the development and application of science,technology and engineering.Basic equations of many modern science are partial differential equations.But most differential equations are difficult to find exact solutions.Therefore,how to solve partial differential equations numerically has become a hot topic.In this thesis,a new numerical method for solving partial differential equations,variablelimit integral method,is studied.The numerical scheme of the initial boundary value problem is designed and solved for Regularization of Long Wave(RLW)equation,Benjamin-BonaMahony-Burgers(BBMB)equation,General Improved KdV(GIKDV)equation and RosenauKdV(RK).The new numerical scheme presents in this thesis is also applicable to other partial differential equations.The specific research content of this paper is as follows.Firstly,Combining with Lagrange three point interpolation function,a new numerical scheme for RLW equation is proposed by using the variational bound integral method.We prove the conservation of energy,the existence of numerical scheme solutions,the convergence and stability of numerical scheme.The convergence order and conservation of time and space are verified by numerical experiments.Secondly,Taylor's fitting function is used as the approximation function,a method for constructing the numerical scheme of second order partial differential equations is given by using the variable-limit integral method.For BBMB equation,construct a new numerical scheme with fourth order in space and second order in time,and prove the existence and uniqueness of numerical solution.In numerical experiments,the error,order of convergence and conserved quantity of different waveform are solved.By comparing with other literature under the same parameters,we know the numerical solution of the numerical scheme constructed in this paper has the advantages of small error,high order of convergence and stable value of the conserved quantity.Thirdly,Taylor's fitting function is used as the approximation function,a method for constructing the numerical scheme of third order partial differential equations is given by using the variable-limit integral method.For GIKDV equation,construct a new numerical scheme with fourth order in space and second order in time.In numerical experiments,the error,order of convergence and conserved quantity of single wave,double wave and three wave are solved.Lastly,Taylor's fitting function is used as the approximation function,a method for constructing the numerical scheme of fourth order partial differential equations is given by using the variable-limit integral method.For Rk equation,construct a new numerical scheme with fourth order in space and second order in time.In numerical experiments,the error,order of convergence and conserved quantity with different parameters are solved.By comparing with other literatures under the same parameters,it can be seen that the error of the numerical solution is small and the conserved quantity is better.