Basins of attraction and the time-dependent approach to obtaining steady-state numerical solutions

Abstract

Abstract The symbiotic relationship between the strong dependence on initial data and the permissibility of spurious stable and unstable asymptotic numerical solutions for commonly used numerical algorithms in computational fluid dynamics (CFD) is illustrated for nonlinear model ordinary differential equations (ODEs) and partial differential equations (PDEs) with known analytic solutions. This is an attempt to understand the global asymptotic behavior of finite difference methods for highly nonlinear and stiff differential equations (DEs). With the aid of a parallel Connection Machine (CM2), the complex behavior and sometimes fractal like structure of the associated numerical basins of attraction of the commonly used time and spatial discretizations are revealed and compared. Here a basin of attraction is a domain of a set of initial conditions whose solution curves (trajectories) all approach the same asymptotic state. Also ¼’e use the term “spurious asymptotic numerical solutions” to mean asymptotic solutions that satisfy the discretized counterparts but do not satisfy the underlying ODEs or PDEs (e.g., nonphysical solutions, nonconvergent solutions or divergent solutions). Asymptotic solutions here include steady-state solutions, periodic solutions, limit cycles, chaos and strange attractors. The results of this investigation can be used as an explanation of the possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.