1993/11/06 by Paul A. Kimoto, Kimoto, Paul A., Chernoff, David F.
Earth and Planetary Sciences · Engineering · #Aquatic and Environmental Studies #Arctic and Antarctic ice dynamics #Astrophysics (astro-ph) #Computational Fluid Dynamics and Aerodynamics #FOS: Physical sciences
paper · pdf · doi:10.48550/arxiv.astro-ph/9311019
openalex publication_date 1993/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate asymptotic convergence in the~Δx → 0 limit as a tool for determining whether numerical computations involving shocks are accurate. We use one-dimensional operator-split finite-difference schemes for hydrodynamics with a von Neumann artificial viscosity. An internal-energy scheme converges to demonstrably wrong solutions. We associate this failure with the presence of discontinuities in the limiting solution. Our extension of the Lax-Wendroff theorem guarantees that certain conservative, operator-split schemes converge to the correct continuum solution. For such a total-energy scheme applied to the formation of a single shock, convergence of a Cauchy error approaches the expected rate slowly. We relate this slowness to the effect of varying diffusion, due to varying linear artificial-viscous length, on small-amplitude waves. In an appendix we discuss the scaling of shock-transition regions with viscous lengths, and exhibit several difficulties for attempts to make extrapolations.