First- and second-order accurate implicit difference schemes for the numerical resolution of the generalized Charney–Obukhov and Hasegawa–Mima equations

Abstract

First- and second-order accurate implicit difference schemes for the numerical solution of the nonlinear generalized Charney-Obukhov and Hasegawa-Mima equations with scalar nonlinearity are constructed. On the basis of numerical calculations accomplished by means of these schemes, the dynamics of two-dimensional nonlinear solitary vortical structures are studied. The problem of stability for the first-order accurate semi-discrete scheme is investigated. The dynamic relation between solutions of the generalized Charney-Obukhov and Hasegawa Mima equations is established. It is shown that, contrary to existing opinion, the scalar nonlinearity in the ease of the generalized Hasegawa-Mima equation develops monopolar anticyclone, while in case of the generalized Charnev -Obukhov equation it develops monopolar cyclone.