Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity.

摘要

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 community observed lack of conservation in SGH methods.