The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics

Abstract

This work explores the somewhat subtle meaning and consequences of the salient properties of the discrete, compatible formulation of Lagrangian hydrodynamics. In particular, since this formulation preserves total energy to roundoff error, the amount of error in the conservation of total energy cannot be used to gauge the internal consistency of calculations, as is often done with the older forms of this algorithm. However, the compatible formulation utilizes two definitions of zone volume: the first is the usual definition whereby the volume of a zone is defined as some prescribed function of the coordinates of the points that define it; the second is given as the integration in time of the continuity equation for zone volume as expressed in Lagrangian form. It is the use of this latter volume in the specific internal energy equation that enables total energy to be exactly conserved. These two volume definitions are generally not precisely equal. It is the analysis of this difference that forms the first part of this study. It is shown that this difference in zone volumes can be used to construct a practical internal consistency measure that not only takes the place of the lack of total energy conservation of the older forms of Lagrangian hydrodynamics, but is more general in that it can be defined on a single zone basis. It can also be used to ascertain the underlying spatial and temporal order of accuracy of any given set of calculations. The difference in these two definitions of zone volume may be interpreted as a type of entropy error. However, this entropy error is found to be significant only when a given calculation becomes numerically unstable, otherwise it remains at or far beneath truncation error levels. In fact, it can be utilized to provide an upper bound on the size of the spatial truncation error for a stable computation. It is also shown how this volume difference can be used as an indicator of numerical difficulties, since exact local conservation of total energy does not guarantee numerical stability or the quality of any numerical calculation. The discrete, compatible formulation of Lagrangian hydrodynamics utilizes a two level predictor/corrector-type of time integration scheme; a stability analysis, both analytical and numerical, is given. This analysis reveals a novel stability diagram that has not been heretofore published, and gives definitive information as to how the stabilizing corrector step should be centered in time.