The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy

• Published in: Journal of computational physics 1998-10, Vol.146 (1), p.227-262
• Main Authors: Caramana, E.J., Burton, D.E., Shashkov, M.J., Whalen, P.P.
• Format: Article
• Language: English
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1006/jcph.1998.6029

The principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought. However, in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost, or obeyed only to an approximation that often becomes worse with time. This is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system. Although a finite truncation error is inherent to all discretization methods, it is possible to satisfy certain global properties, such as conservation of mass, momentum, and total energy, to numerical roundoff error. The purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties. In particular, it is shown how conservation of total energy can be utilized as an intermediate 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 other 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 viscosity or subzonal pressures. These forces originate only in discrete form; nonetheless, the change in internal energy caused by them is still completely determined. The procedure given here is compared to the [method of support operators,] to which it is closely related. Difficulties with conservation of momentum, volume, and entropy are also discussed. The proper treatment of boundary conditions and differencing with respect to time are detailed.