## 2. Numerical methodThe code used is temporally and spatially second order accurate and takes account of cooling due to ionisation and radiative atomic transitions. It uses a MUSCL-type scheme (e.g. van Leer 1977) to integrate the hydrodynamic and ionisation equations. The advection terms are calculated in a straightforward upwind fashion, while all other differences are centered. The details and tests of this code are fully described in Downes (1996). The simulations are purely hydrodynamic and are performed in slab symmetry. ## 2.1. Equationswhere is the mass density,
is the fluid velocity, We assume the gas law to be where where is the heat capacity of the gas. The
energy loss, where is the energy loss due to radiative
atomic transitions, is the ionisation potential
of atomic hydrogen, and The rate of ionisation of atomic hydrogen Here and are the
collisional ionisation and radiative recombination coefficients in
cm ## 2.2. SchemeAs an example of how we integrate Eqs. 1to 5let us consider a continuity equation for an arbitrary variable . The scheme we use is a two-step scheme which steps to time from using an upwind first order scheme. The values of the variables at time are then used to calculate fluxes which are second order accurate in space. These fluxes are used to integrate from time to time . This process leads to a scheme which is second order in time. Specifically, we define , the first order flux across the cell boundary at time , as where and is the advection velocity. To calculate
the second order flux at time we make an
estimate of the gradient, , of the variable
within cell where the function av is a non-linear averaging function given by (van Leer 1977) We can now define the second order flux at and time as Note that the advection velocity itself cannot easily be calculated in an upwind fashion and is approximated by A temporally and spatially second order accurate scheme is then given by first solving where . Then, using the values to calculate , we solve the equation ## 2.3. The cooling and heating functions
The cooling function used in this work is a combination of that used
by Rossi et al. (1997) and that contained in Sutherland & Dopita
(1993). At low temperatures () it is probable
that the former is more accurate. However at higher temperatures this
cooling function fails as it only calculates cooling due to at most
singly ionised atomic species. The latter cooling function is
calculated for a gas of cosmic abundances cooling from about
K and it does not take account of
non-equilibrium ionisation (unlike the Rossi et al. (1997) cooling
function). The resulting function is plotted in Fig. 1. Tests of
our implementation of the cooling have shown that it performs very
well, predicting reasonably accurately the stability limit as well as
the amplitude of oscillation of an overstable radiative shock wave
(see Downes 1996). Note that this test is for rather strong shocks
( km s
Rossi et al. (1997) also used a constant volume heating function which cancels the cooling function at the equilibrium temperature, . Here we compare the results of simulations which use this heating term with those which simply assume that, below ( K in this work) the cooling is insignificant and therefore can be set to zero. Rather than simply cutting off the cooling at we attenuate the cooling function shown in Fig. 1 as follows: This effectively implies that the cooling is zero at K. Note that this means that the cooling function is a very shallow function of temperature around . The assumption of insignificant cooling below is often used in simulations of purely atomic YSO jets as, below K, the atomic cooling rate falls off rapidly (Blondin et al. 1990). We find that these two approaches to maintaining equilibrium in the initial configuration produce significantly different results. In addition we have run simulations where the heating term is proportional to the density, as might be the case if radiative transfer was occurring. © European Southern Observatory (ESO) 1998 Online publication: March 3, 1998 |