2000/03/23 by A. J. Roberts, Roberts, A. J.
Earth and Planetary Sciences · Engineering · #37L10 #37L65 #37M99 #65M20 #65P40 #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Meteorological Phenomena and Simulations #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.math/0003135
openalex publication_date 2000/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
I prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small. Using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing. For example, the advection-diffusion equation is found to be stably modelled for all advection speeds and all grid spacing. The theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory. When numerically solving nonlinear partial differential equations, this approach can be used to derive systematically finite difference models which automatically have excellent characteristics. Their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems.