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 were made among systems of equations with timing modifications. These comparisons led to experimentally superior differencing forms. Stability analyses of these difference forms showed the reasons for the superiority of one form over another. Of greater importance, the stability criteria obtained showed the function of an artificially introduced diffusion term required in the treatment given to shocks. The stability criterion in each case involved the familiar Courant condition and a term which corresponds to the stability criterion of the diffusion equation. Upper limits to the magnitude of the ccefficient of the diffusion firm were established as a function of Courant number. While lower limits were also indicated, they required modification when shocks were involved. Alternate differencing schemes were considered in which the previously-used total energy calculation was 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 were manifested in distortions of profiles at material discontinuities through which shocks passed and in rarefactions associated with such occurrences, The errors in turn effected stability in the vicinity of the material discontinuities. (auth)