Up: A nonlinear convective model
Subsections
The equations of convective radiation hydrodynamics given in Table 2 yield a
system of 7 nonlinear coupled partial differential equations. Together with the initial
and boundary condition discussed in Sects. 2.8 and
2.7 we obtain a initial-boundary-value problem which is solved
numerically. The basic features of the employed numerical method are
compiled in the following section, while detailed discussions can be found
in the quoted papers.
For a continuous resolution and tracking of the driving zones and other moving flow
features during the pulsation cycle we employ the adaptive grid algorithm developed by
Dorfi & Drury (1987). The basic concept is to distribute the
available spatial grid points uniformly in arc-length along the graphs of
the important physical variables. This is done by solving an additional
equation (i.e. the adaptive grid equation) simultaneously with the physical
equations. The major input to the grid equation is a set of physical
quantities named marker functions, which control the grid distribution.
This set has to be chosen according to the investigated problem. For
stellar pulsations the set of marker functions given in Table 5
turned out to be appropriate for resolving and tracking all important
physical quantities. In particular we found the isentropic temperature
gradient 
to be very useful in order to resolve the HeII ionization
zone responsible for the driving in Cepheids and RR Lyrae stars. Each
marker function can be weighted separately in order to account for
different scales and the corresponding weight factors are also given in
Table 5. Further information about the adaptive grid equation in
the context of stellar pulsation calculations can be found in
Feuchtinger & Dorfi (1994, 1997).
Table 5:
Marker functions and weights adopted in the adaptive grid equation for the
case of classical stellar pulsations
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By adding the adaptive grid equation to the CRHD system (cf. Table 2) we end
up with a system of 8 nonlinear partial
differential equations. These equations are discretized in space and time by the method
given in Tscharnutter & Winkler (1979) and Winkler & Norman
(1986), which implies that
the equations are formulated in conservation form. The employed volume
discretization guarantees that global conservation of quantities like mass, momentum or
internal energy, as prescribed by the differential equations, becomes an analytical
property of the discretized equations.
Advection terms are treated by the monotonic scheme of van Leer
(1977) which is 
order in space. For the discretization
of the time derivatives we adopt order
centered differences which requires an implicit method of solution (Winkler
& Norman 1986).
The resulting nonlinear algebraic system of discretized equations is solved implicitly
by means of a multidimensional Newton-Raphson method. It is important to note that
in contrast to explicit solution methods the timestep of the numerical solution is
determined only by accuracy considerations and is not explicitly limited by extra
relations
like the Courrant-Friedrichs-Lewy condition. For further details we refer to Dorfi &
Feuchtinger (1995).
In order to treat shock waves and other discontinuities by the finite difference method
reviewed in the last section we use the artificial tensor viscosity developed by
Tscharnutter & Winkler (1979). In this context it is important
to guarantee that the physical properties of the model are independent of
the artificial viscosity. This requires the respective viscous length scale
to be lower than the typical physical length scales of the problem. In the
case of classically pulsating stars the important length scale is the
pressure scale hight of the hydrostatic background structure which in the
relevant regions is of the order of 10-1 - 10-3 times the local
radius (depending on the particular type of star and the position within the
envelope). Therefore we chose a viscous length scale of the order 10-4
times the local radius in order to prevent that the results depend on the artificial viscosity.
The radiative models in Feuchtinger & Dorfi (1994) use an
additional artificial viscosity to adjust the amplitude of the pulsations.
In the case of our convective model this viscosity is replaced by the
turbulent eddy viscosity which serves as a free but physically motivated
parameter and is used to tune the pulsation amplitudes.
Table 6:
List of symbols
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Up: A nonlinear convective model
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