Research On The Construction Method Of Numerical Scheme With Conservation

Partial differential equations(PDEs)play an important role in the development of contemporary science and technology and in solving engineering problems.In practical engineering applications,it is very difficult to solve most partial differential equations.Therefore,in order to conduct practical research,it is necessary to construct a corresponding numerical format to find its analytical solution.In this thesis,we mainly study the construction of numerical schemes with conservation using variable limit integral method.The main idea of the integral method with variational limit is to integrate the original differential equation many times,so that the partial differential equation can be transformed into an integral equation without the derivative term of the unknown function,it will be convenient to construct a discrete scheme for approximate solution of PDEs by fitting unknown functions.In this thesis,we use the integral method with variational limit,combined with the Lagrange interpolation function as an approximation function.Taking the highest derivative of the equation for space as the second,third,and fourth orders as examples,we give a method of how to determine the parameters of the integration limit under the specified accuracy in order to construct a conservative numerical scheme.The improved Regularized Long Wave(MRLW)equation,Kd V equation and Rosenau-Kd V-RLW(RKR)equation are constructed and solved by this method.Firstly,a new conservation numerical scheme is constructed for the non-linear MRLW equation with the second derivative of the highest order for space.The existence of numerical solution,the conservation,convergence and stability of numerical schemes are strictly proved.In addition,it is numerically verified that both the temporal and spatial convergence order of the proposed scheme are second order,and that the energy of the discrete form is conserved.Secondly,a new conservation numerical scheme is constructed for the non-linear Kd V equation with the third derivative of the highest order for space.The existence of the solution and the conservation of the numerical scheme are strictly proved.In addition,it is numerically verified that both the temporal and spatial convergence order of the proposed scheme are second order,and that the energy of the discrete form is conserved.Finally,a new numerical scheme for conservation is constructed for the nonlinear RKR equation,which the highest derivative in space is the fourth order.This scheme achieves second-order accuracy in space and time.The existence and uniqueness of numerical solutions,the conservation,convergence and stability of numerical schemes are strictly proved.