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Astron. Astrophys. 345, 977-985 (1999)

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Appendix A: overcoming negative pressures

As noted in Sect. 2.1 it is common for conservative numerical codes to predict negative pressures under certain conditions. Flows giving rise to such problems are usually highly supersonic, and diverging. The difficulties occur because the ratio of internal to total energy goes like [FORMULA], where M is the Mach number of the flow. Therefore, in a highly supersonic flow, the fractional error required to predict a negative pressure is rather small. The introduction of energy losses exacerbates this problem. This is due to the fact that relatively weakly diverging flows, for example, can become supersonically diverging when the system is cooled because the sound speed is reduced.

Schemes which are second order accurate in space tend to produce more negative pressures than first order ones. This is because first order schemes dissipate strong features quickly so that strongly diverging flows rarely occur. It seems reasonable, then, to invoke a scheme which is first order in space whenever a negative pressure is produced as this will introduce extra dissipation. This extra dissipation may eliminate the negative pressure by moving some extra energy from neighbouring cells into the problem one. The scheme used by the code in this work can be summarised as follows:


where [FORMULA] and [FORMULA] are the state vector and second order flux calculated at [FORMULA] and [FORMULA] respectively, and [FORMULA]. If this scheme produces a negative pressure at [FORMULA] then simply apply Eq. A1 using the first order fluxes [FORMULA] instead of the second order ones. Note that it is necessary to adjust the neighbouring cells also so that overall conservation is maintained.

While this fix does not work for all problems, it was found to be rather effective in the simulations presented here where very strong rarefactions are produced both at the edge of the jet at the [FORMULA] boundary, and within the jet itself where the enforced velocity variations can cause problems. This fix can be implemented very easily by ensuring that the flux vectors around problem cells are stored. It has the advantage that it does not involve losing conservation of any of the physically conserved quantities. Reducing the scheme to first order in certain regions of the grid is not a significant problem because this happens around shocks anyway in order to maintain monotonicity.

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© European Southern Observatory (ESO) 1999

Online publication: April 28, 1999