Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity
For general formulations of staggered-grid hydrodynamics (SGH), we show that exact difference expressions for total energy conservation are derivable directly from the difference expressions for nodal mass and momentum conservation. The results are multi-dimensional and apply to arbitrarily (as well as regularly) connected grids. In SGH the spatial centering of coordinates and velocity are at nodes while mass, internal energy, and pressure are within zones; temporally, variables are usually centered at the integer time step except for velocity which is at the half step.
The momentum equation is found to lead to an exactly conserved energy flux between internal energy in the zones and kinetic energy at the nodes. Exact expressions for work, kinetic and internal energy evolution, and total energy conservation follow. The derived work expression also shows that momentum, kinetic and internal energies can be exactly defined at either full or half time steps. The energy flux is not properly calculated by those SGH methods in which the work is differenced independently of the differenced momentum equation. This leads to the commonly observed lack of conservation in SGH methods.
KeywordsInternal Energy Momentum Equation Blast Wave Lawrence Livermore National Laboratory Nodal Mass
Unable to display preview. Download preview PDF.
- Amsden, A.A. and Hirt, C.W. (1973), “YAQUI: An Arbitrary Lagrangian-Eulerian Computer Program for Fluid Flow at All Speeds,” Los Alamos National Laboratory, Report LA-5100.Google Scholar
- Crowley, W.P. (1970), “FLAG: A Free-Lagrange Method for Numerically Simulating Hydrodynamic Flows in Two Dimensions,” Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics, Springer-Verlag, pp. 37–43.Google Scholar
- Fritts, M.J., Crowley, W.P., and Trease, H. (1985), The Free Lagrange Method, Springer-Verlag, New York.Google Scholar
- Godunov, S.K. (1959), Mat. Sb. 47, p. 271.Google Scholar
- Taylor, G.I. (1941), British Report RC-210.Google Scholar
- Taylor, G.I. (1950), “The formation of a blast wave by a very intense explosion. The atomic explosion of 1945,” Proc. Roy. Soc. (London), Ser. A, 201, pp. 175–186.Google Scholar
- Trulio, John G., and Trigger, Kenneth R. (1961a), “Numerical Solution of the One-Dimensional Lagrangian Hydrodynamic Equations,” Lawrence Livermore National Laboratory, Report UCRL-6267.Google Scholar
- Trulio, John G., and Trigger, Kenneth R. (1961b), “Numerical Solution of the One-Dimensional Hydrodynamic Equations in an Arbitrary Time-Dependent Coordinate System,” Lawrence Livermore National Laboratory, Report UCRL-6522.Google Scholar
- von Neumann, J. (1941), NDRC, Div. B, Report AM-9.Google Scholar
- von Neumann, J. (1950), “The Point Source Solution,” in Los Alamos National Laboratory, Report, LA-2000.Google Scholar
- von Neumann, J., and Richtmyer, R. (1950), “A Method for the Numerical Calculation of Hydrodynamic Shocks,” J. Appl. Phys. 21, p. 232.Google Scholar
- Youngs, D.L. (1978 ), Atomic Weapons Establishment, private communication.Google Scholar
- Zel'dovich, Y.B., and Raizer, Y.P. (1966), Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Academic Press, New York.Google Scholar