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**The construction of compatible hydrodynamics algorithms utilizing conservation of total energy.**
*(English)*
Zbl 0931.76080

The purpose of this work is to show how the hydrodynamic equations can be differenced compatibly so that they obey the conservation 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, together with the proper treatment of boundary conditions and differencing with respect to time. \(\copyright\) Academic Press.

### MSC:

76M99 | Basic methods in fluid mechanics |

76M20 | Finite difference methods applied to problems in fluid mechanics |

### Keywords:

Lagrangian form of hydrodynamic equations; Sedov blast wave problem; conservation of volume; conservation of entropy; conservation of mass; method of support operators; staggered spatial variables; momentum equation; internal energy equation; conservation of momentum; boundary conditions; differencing
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\textit{E. J. Caramana} et al., J. Comput. Phys. 146, No. 1, 227--262 (1998; Zbl 0931.76080)

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