Research On High-accurate Structure-preserving Algorithms For The Rosenau Equation

Rosenau equation is an important of nonlinear partial differential equations,which is typically used to describe various complex nonlinear wave phenomena and attracts a lot of interest from scholars at home and abroad.In this thesis,based on the basic idea of the structure-preserving algorithms,we first propose highaccurate momentum-preserving and energy-preserving algorithms for the Rosenau equation,respectively.The original equation is rewritten as a self-adjoin system.Then symplectic Runge-Kutta method in time and the standard Fourier pseudo-spectral method in space are employed to discretize the resulting system,respectively,and a fully discrete high accurate momentum-preserving scheme is obtained.For the derivation of the energy-preserving scheme,the original system is reformulated into an equivalent form with quadratic energy by introducing one or more quadratic auxiliary variables.We then apply the symplectic Runge-Kutta method in the time and the standard Fourier pseudo-spectral method in the space to discretize the reformulated system,respectively.We show that the proposed scheme can conserve the original energy and mass.For the implicit scheme,a nonlinear equation shall be solved by using a nonlinear iterative method at each time step,and thus it may be time consuming.To improve the computational efficiency,based on the linearized strategies,such as predictor-corrector approach and so on,the symplectic Runge-Kutta method as well as the standard Fourier pseudo-spectral method,we propose a class of high-accurate,linearly implicit and momentum-preserving schemes for Rosenau equation.Finally,Numerical results are addressed to demonstrate the accuracy and efficiency of the proposed scheme.