Numerical Stability of the Fast Convergent Diagonally Approximating Difference Schemes

摘要

A general nonlinear method for the uniform approximation of sufficiently smooth solutions of nonlinear evolution partial differential equations (PDEs) by solutions of specially constructed sequences of ordinary differential equations (ODEs) is discussed. Computer applications showed that explicit one-step numerical approximations of the solutions of the ODEs exhibited fast convergence to the smooth solution of the initial evolution PDEs. The stability of the fast convergent diagnoal approximation in the presence of round-off errors is proved. For the sake of simplicity the proof is given for the linear case of a pure initial value problem with data in a certain class of quasi-analytic functions. It is also possible, but more difficult, to prove stability in the nonlinear case and for a wider classes of initial data.