THE CONSTRUCTION OF COMPATIBLE HYDRODYNAMICS ALGORITHMS UTILIZING CONSERVATION OF TOTAL-ENERGY

The principal goal of all numerical algorithms is to represent as fait hfully and accurately as possible the underlying continuum equations t o which a numerical solution is sought. However, in the transformation of the equations of fluid dynamics into discretized form important ph ysical properties an either lost, or obeyed only to an approximation t hat often becomes worse with time, This is because the numerical metho ds used to form the discrete analog of these equations may only repres ent them to some order of local truncation error without explicit rega rd to global properties of the continuum system. Although a finite tru ncation error is inherent to all discretization methods, it is possibl e to satisfy certain global properties, such as conservation of mass, momentum, and total energy, to numerical roundoff error. The purpose o f this work is to show how these equations can be differenced compatib ly so that they obey the aforementioned properties. In particular, it is shown how conservation of total energy can be utilized as an interm ediate device to achieve this goal for the equations of fluid dynamics written in Lagrangian form, and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system. For staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each oth er in a simple and generic manner by use of the conservation of total energy, This allows for the specification of forces that can be of an arbitrary complexity, such as those derived from an artificial viscosi ty or subzonal pressures. These forces originate only in discrete form ; nonetheless, the change in internal energy caused by them is still c ompletely determined. The procedure given here is compared to the ''me thod of support operators,'' to which it is closely related. Difficult ies with conservation of momentum, volume, and entropy are also discus sed. The proper treatment of boundary conditions and differencing with respect to time are detailed. (C) 1998 Academic Press.