Application of nonlinear dynamics approaches to the numerical study of PDE

Abstract

In this paper we analyze the structure of dynamical systems which are produced by discretizing the nonlinear partial differential equations (PDEs) from some viewpoints such as analytical and/or numerical approaches and qualitative and/or quantitative ones. A 1D Burgers' equation is selected as a simple model, and the structures of the numerical asymptotic solutions of the finite difference equations which are given by discretizing the original DE are considered. The dependence of the equation form and the dependence of the scheme on the nonlinear structure of the numerical solutions are discussed. Furthermore, we apply these analyses to a practical CFD calculation. The dimensions of attractors constructed from the time series of one of the variables in the numerical results, which were given by solving the Navier-Stokes equations directly, are calculated. The instability of the shear boundary in the mixing process is discussed. In these analyses, the dependence of the discretization parameter and the grid system on the steady-state numerical solutions is also discussed. (Author)