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Convergence properties of finite-difference hydrodynamics schemes in the presence of shocksWe investigate the asymptotic convergence of finite-difference schemes for the Euler equations when the limiting solution contains shocks. The Lax-Wendroff theorem guarantees that certain conservative schemes converge to correct, physically valid solutions. We focus on two one-dimensional operator-split schemes with explicit artificial-viscosity terms. One, an internal-energy scheme, does not satisfy the assumptions of Lax-Wendroff; the other, a conservative total-energy scheme, does. With viscous lengths chosen proportional to the grid size, we find that both schemes converge to their zero-grid-size limits at the theoretically expected rate, but only the conversative scheme converges toward correct solutions of the inviscid fluid equations. We show that the difference in their behaviors results directly from the presence of shocks in the limiting solution. Empirically, we find that when the viscous lenghts tend toward zero more slowly than the grid size, however the nonconservative scheme also converges toward correct solutions. We characterize the asymptotic behavior of the total-energy scheme in a particular problem in which a shock forms. As the grid is refined, a Cauchy error approaches the expected rate of change slowly. We show that the changes in the artificial viscosity alter the diffusion of small-amplitude waves. The differences associated with such waves make the dominant contribution to the Cauchy error. We formulate an analytic model to relate the rate of approach to the effect of varying diffusion in waves and find quantitative agreement with our numerical results.
Document ID
19950045460
Acquisition Source
Legacy CDMS
Document Type
Reprint (Version printed in journal)
External Source(s)
Authors
Kimoto, Paul A.
(Cornell Univ. Ithaca, NY, United States)
Chernoff, David F.
(Cornell Univ. Ithaca, NY, United States)
Date Acquired
August 16, 2013
Publication Date
February 1, 1995
Publication Information
Publication: The Astrophysical Journal Supplement Series
Volume: 96
Issue: 2
ISSN: 0067-0049
Subject Category
Fluid Mechanics And Heat Transfer
Accession Number
95A77059
Funding Number(s)
CONTRACT_GRANT: NAGW-2224
CONTRACT_GRANT: NSF AST-86-57467
Distribution Limits
Public
Copyright
Other

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