Study On Numerical Difference Schemes And Computational Stability Of Nonlinear Evolution Equations

In this paper, the computational stability of the two-dimensional original nonlinear equation from atmospheric dynamics is studied, by the Hirt heuristic analysis. It is proved through theoretical analysis and numerical tests that the computational stability of difference schemes is not only depend on the structure of scheme, but also on the form of initial values and their partial derivatives, and the necessary conditions of computational stability are given. Similarly, the computational stability of nonlinear Schrodinger equation is analyzed, and the necessary condition of computational stability is given. The result indicates that the close relationship between the computational stability of the difference schemes and the properties of the solution are revealed. Then, the stability of the numerical calculation for the coupled nonlinear Schrodinger equation is further discussed. The format of four-point implicit scheme for the coupled nonlinear Schrodinger equations is studied. It is found that if the conditions satisfy A_j0∈H¹, 0≤b≤|a_j|≤M, and 0≤q_j(s₁,s₂)≤R(s₁+s₂), s_j∈[0,∞), the format of four-point implicit scheme is proved to be stable. Furthermore the error is O(Ï„+h²).