For the case of classical RR Lyrae pulsations it has already been shown, that long standing problems like the Fourier phase discrepancy or the computation of stable double mode pulsations can be resolved by using the outlined method (Feuchtinger & Dorfi 1997; Feuchtinger & Dorfi 1998; Feuchtinger 1998). An application to radiative Cepheid pulsation has been demonstrated in Dorfi & Feuchtinger (1995), even though further computations are necessary in order to corroborate the applicability to these objects.
Due to the high spatial resolution provided by the adaptive grid it is possible to include the stellar atmosphere into the pulsation model. In this case the oscillations of the underlying envelope induce high amplitude shock waves propagating down the steep density gradient. For classical stellar pulsations it is generally assumed that the influence of the atmosphere can be neglected and this assumption was corroborated by a nonlinear RR Lyrae model including the atmosphere in Feuchtinger & Dorfi (1994). Further applications of the model including detailed investigations of Cepheids and RR Lyrae stars are in progress and will be published in forthcoming papers.
One major unsolved problem in stellar pulsation theory is the computation of AGB pulsations (e.g. Tuchman 1998; Höfner 1998). In contrast to classically pulsating stars characterized by weak nonadiabaticity and small convection zones, AGB envelopes are dominated by convective energy transfer and highly nonadiabatic pulsations. Consequently a nonvanishing amount of mechanical energy is deposited into the stellar atmosphere which in turn influences the pulsating envelope. The levitation of the atmosphere through the pulsations yields conditions where molecules and dust grains can form and the radiation pressure on these new opacity sources causes a stellar wind to develop (e.g. Fleischer et al. 1992).
Consequently a pulsation model applicable to AGB stars must be capable of including the stellar atmosphere with its complicated physical processes like time-dependent dust formation, propagating shock fronts and radiation pressure driven winds. That this is possible with the method presented in this paper has been demonstrated by Höfner & Dorfi (1997) who calculated the detailed structure of AGB atmospheres where the pulsations of the underlying star is simulated by a moving piston.
The remaining problem is to compute these pulsations, which mainly seems to be caused
by an accuracy deficit. In contrast to classical pulsations AGB stars are
gravitationally
weakly bound. Energetical considerations reveal the total energy of AGB stars to be
small compared with the internal energy, while e.g. for Cepheids or RR Lyrae stars the
total energy always is of the order of the internal energy. Additionally the mass
weighted mean of 
taken over a typical AGB envelope lies only slightly above
the critical value of 4/3 indicating dynamical instability. Numerically speaking
small errors in the internal energy as produced by discretization or advection errors
can be quite large when compared with the total energy. An amplification of such errors
finally leads to a dynamical instability caused by numerical errors. As a consequence
the main problem is to solve the nonlinear equations with sufficient accuracy.
Finally we want to draw attention to completely different astrophysical problems like evolution of supernova remnants (Dorfi 1994), formation of gaseous planets (Wuchterl 1993) and protostellar collapse calculations (Wuchterl 1998), where the same physical and numerical philosophy as reviewed in this paper has been employed successfully.
Acknowledgements
This work is supported by the Fonds zur Förderung der wissenschaftlichen Forschung (FWF) under project number S7305-AST.
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