Lagrangian Difference Approximations for Fluid Dynamics
Abstract
Various procedures are given for writing explicit difference approximations to the one-dimensional lagrangian hydrodynamics equations. Computational comparisons are made among systems of equations with timing modifications. These comparisons lead to experimentally superior differencing forms. Stability analyses of these difference forms show the reasons for the superiority of one form over another. Of greater importance, the stability criteria obtained show the function of an artificially introduced diffusion term required in the treatment here given to shocks. The stability criterion in each case involves the familiar Courant condition and a term which corresponds to the stability criterion of the diffusion equation. Upper limits to the magnitude of the coefficient of the diffusion term are established as a function of Courant number. While lower limits are also indicated, they require modification when shocks are involved. Alternate differencing schemes are considered in which the previously-used total energy calculation is replaced by an internal energy calculation. It is shown that care must be taken that the kinetic and internal energies are expressible in terms of local quantities. That is, in addition to the equations being conservative in a gross sense, they must also be locally conservative. This is necessary in order that the energy condition of the Rankine-Hugoniot equations be satisfied when shocks arise. Finally discussion is given to errors resulting from the replacement of shocks by a shock layer, that is, errors connected with the artificially inserted diffusion term. These errors are manifested in distortions of profiles at material discontinuities through which shocks have passed and in rarefactions associated with such occurrences. The errors in turn effect stability in the vicinity of the material discontinuities.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jun 14, 1961
- Accession Number
- ADA382488
Entities
People
- Jacob E. Fromm
Organizations
- Los Alamos National Laboratory