Research On High-accuracy Algorithm For Two Types Of Shallow Water Wave Equations

In modern science and engineering,a large number of mathematical models are described by partial differential equations.Therefore,solving them has become a major task of scientific and engineering calculations.However,it is very difficult to solve the exact solution of partial differential equations.So,we consider to replace the exact solution approximately by the numerical solution.Obtaining the numerical solution of partial differential equations has attracted extensive attention of scholars at home and abroad.In this paper,we mainly study the numerical solutions of the generalized RosenauKawahara-RLW equation and Camassa-Holm equation.The processing methods are as followed.Firstly,we introduce auxiliary variables to reduce the orders of the original equation,and use the Variable Limit Integral method to construct its numerical schemes for the equation with the initial and boundary conditions;Secondly,the existence and uniqueness of the numerical solution and the energy conservation,convergence and stability of the numerical schemes are all proved;Finally,numerical experiments are carried out.During this process,we estimate the errors between the numerical solution and the analytical solution.At the same time,we verify the effectiveness of the numerical schemes.In the first part,we study the generalized Rosenau-Kawahara-RLW equation.Firstly,reduce the orders of the original equation.That is to introduce the auxiliary variables to reduce the fifth partial derivative of space direction to the second partial derivative.And combine the Variable Limit Integral method with Taylor function method to discrete the space direction.So we obtain the semi-discrete numerical schemes.Secondly,use the Crank-Nicholson method to discrete in the time direction and obtain the two-level nonlinear implicit fully-discrete schemes.The numerical schemes have fourth order accuracy in space direction and second order accuracy in time direction.The second work is to prove the existence and uniqueness of the numerical solution and the energy conservation,convergence and stability of the numerical schemes.Finally,carry out numerical experiments to prove the effectiveness of the numerical schemes with analyzing the energy property of the numerical schemes and the errors between analytical solution and numerical solution.In the second part,we study the Camassa-Holm equation,which reflects the problem of shallow water waves.Firstly,introduce the auxiliary variables to establish the numerical schemes.In space direction,the semi-discrete schemes are obtained by combining the Variable Limit Integral method and the Taylor function method.In time direction,we use the CrankNicolson method to obtain the nonlinear implicit fully-discrete schemes with the accuracy of fourth order and second order in space and time directions,respectively.Secondly,prove the existence and uniqueness of the numerical solution and the energy conservation,convergence and stability of the numerical schemes.Finally,carry out numerical experiments to prove the effectiveness of the numerical schemes.