High-order Accurate Finite Difference Schemes For The Several Classes Of Nonlinear Partial Differential Equations

Numerical methods for nonlinear partial differential equations have been widely used in the modern field of science and engineering,most numerical methods have low convergent order and slow efficiency,etc.,which cannot meet actual application.The research of high-order accurate scheme is very important in scientific engineering calculation.In this paper,we study the high-order accurate schemes for the generalized Rosenau-Kd V(GRKd V)equation,the damped generalized symmetric regularized long wave(DGSRLW)equations,the symmetric regularized long wave(SRLW)equations and the coupled nonlinear Schr?dinger(CNLS)equations by using the finite difference method.At first,a three-time level linear high-order accurate difference scheme for the GRKd V equation is proposed.Conservation of the scheme,existence and uniquely solvability of the numerical solution,stability and convergence with order O(?~2+h~4)in the L~?-norm of the scheme are proved by the discrete energy method.Numerical examples support the theoretical analysis and the efficiency of the scheme,and it is well applied to solve the Kd V equation.Secondly,we discussed the properties of the solution for the DGSRLW equation.Two high-order accurate difference schemes with two-time level nonlinear coupling and three-time level linear decoupling are proposed.Dissipation of the schemes,existence and uniquely solvability of the numerical solution,stability and convergence with order O(?~2+h~4)in the L~?-norm and the L~2-norm of the scheme are proved by the discrete energy method.An iterative algorithm is designed for the two-time nonlinear coupled scheme and its convergence is proved in detail.Numerical examples studied that the dynamic simulation of soliton-soliton head-on collisions and catch-up collisions and the change in the total energy dissipation of colliding system when different damping parameters.Furthermore,a fourth-time level linear high-order accurate difference scheme for the SRLW equation is proposed.Conservation of the scheme,existence and uniquely solvability of the numerical solution,stability and convergence with order O(?~2+h~4)in the L~?-norm of the scheme are proved by the discrete energy method.Numerical examples are given to verity the conservation,convergence order and stability of the compact scheme and study the dynamic simulation of soliton-soliton head-on collisions and catch-up collisions.Finally,a two-time level nonlinear coupled high-order accurate difference scheme for the CNLS equation is proposed.Conservation of the scheme,existence and uniquely solvability of the numerical solution,stability and convergence with order O(?~2+h~4)in the L~?-norm of the scheme are proved by the discrete energy method.An iterative algorithm is designed for the two-time nonlinear coupled scheme and its convergence is proved in detail.Numerical examples support the theoretical analysis studied three cases of soliton-soliton collision.The results of the simulations agree with those obtained in[51,57,67].