Study Of High Accuracy Compact Difference Scheme For GRKDV Equation And GRB Equation

Many problems in the field of science and engineering can be described by partial differential equations,the most of the accurate solution of partial differential equations with the practical application background can't work out,or analytical expression of solution is complex.The numerical method is widely used.In this paper,we study the initial-boundary value problems of the generalized Rousenau Burgers(GRB)equation and the generalized Rousenau Korteweg de Vries(GRKDV)equation by the difference method.At first,a high accuracy compact difference scheme for the generalized Rousenau Burgers(GRB)equation is proposed.Existence of its numerical solutions with the energy analysis method has been shown,the difference scheme is unconditionally stable and convergence,and its numerical convergence order is O(?2+h4)in the L?-norm.Secondly,a two-layer high accuracy compact and conservative difference scheme for the generalized Rousenau Korteweg de Vries(GRKDV)equation in 2D is proposed.Combining with the energy analysis method,the system meets the conservative laws in energy and mass.Existence of its difference solutions has been shown.The scheme is unconditionally stable and convergence,and its numerical convergence order is O(?2 +H14 +h24)in the L?-norm.Finally,a three-layer high accuracy compact and convergence difference scheme for the generalized Rousenau Korteweg de Vries(GRKDV)equation in 2D is proposed.Combining with the energy analysis method,the system meets the conservative laws in energy and mass.Existence of its difference solutions has been shown.The scheme is unconditionally stable and convergence,and its numerical convergence order isO(?2+H14+h24)in the L?-norm.On the basis of theoretical proof,the numerical experiment for the difference scheme of the proposed has been shown.Numerical results show the validity and reliability of the numerical theory.