Local Structure-preserving Algorithms For The Damped KdV Equation

The idea of structure-preserving algorithm was systematically presented by Feng in the 1980s.Symplectic and multi-symplectic algorithms have shown superior prop-erties in numerical simulation Hamiltonian partial differential equations(PDEs).In recent years,designing structure-preserving algorithms for the damped system attracts a lot of attention and some novel methods,such as the splitting method and the ex-ponential integrator are presented.In this thesis,we focus on discussing the multi-symplectic structure and the structure-preserving properties of the general Hamiltoni-an PDEs with linear damping,and derive three novel structure-preserving algorithms.The first algorithm is obtained by discretizing the damped multi-symplectic formu-lation with symplectic Euler method in time and the midpoint method in space,re-spectively.With the aid of the midpoint method in time and the symplectic Euler method in space,the second algorithm is presented.By replacing the symplectic Eu-ler method with the Fourier pseudospectral method in the second algorithm,the third algorithm is derived.We rigorously prove that the proposed three algorithms satis-fy the conformal multi-symplectic conservation law.Then,the proposed algorithms are employed to solving the damped Korteweg-de Vries equation,where three new structure-preserving scheme are obtained.Furthermore,we also proposed a local con-formal momentum-preserving algorithm for the damped Korteweg-de Vries equation.Numerical experiments are presented to illustrate the effectiveness of the proposed structure-preserving algorithms.