Validation of the scheme

The validation of the numerical scheme adopted is crucial concerning the reliability of the results. One problem concerns the resolution provided by the Lagrangian grid,

-12 -9 -6 -3 0

0.2 0.4 0.6 0.8 1

LogDensity(g/cm3)

Normalized Radius

19 M

60 M

FIGURE2.13: Snapshots of the density stratification as a function of nor-malized radius (normalization by the contemporary maximum radius) for two stellar models during finite amplitude pulsations. Note the

dif-ferent distribution of the 500 grid points used.

in particular, if shock waves are present. For illustration, the distribution of the 500 grid points used is shown in Fig. 2.13for an instance of time during finite amplitude pulsations of two stellar models. For the less massive model, the structure of the star including the shock wave is well resolved. However, for the massive model, two of the three shock waves have attracted almost all grid points, thus being well resolved them-selves but leaving only a few grid points for the third shock and the remaining part of

Chapter 2. Basic equations and methods 27 the envelope. Thus a large part of the model is not sufficiently well resolved, which is a consequence of the Lagrangian grid moving together with the mass elements. To solve the problem, grid points have to be inserted in underresolved regions, i.e., the Lagrangian grid has to be reconstructed. Attempts towards a conservative grid recon-struction procedure have been done by Chernigovski et al. (2004). However, a fully satisfactory grid reconstruction approach is still not available. In the present study we shall restrict ourselves to sufficiently well resolved models obtained with a constant number of grid points.

Another problem concerns the artificial viscosity introduced to smooth the discon-tinuities appearing during the propagation of shocks. To determine the right amount of artificial viscosity is important for the quality of the results. For example, if the arti-ficial viscosity is too high, it can damp the physical instabilities. On the other hand, if it is too small, numerical oscillations around the shock will occur (Gibb’s phenomenon), and finally the shock waves can no longer be handled. Therefore, in order to avoid numerical artifacts, the artificial viscosity has to be chosen with care.

4.3 hours

FIGURE2.14: Simulation of the evolution of an instability for a 90 M

ZAMS model from hydrostatic equilibrium into the nonlinear regime.

The photospheric velocity is given as a function of time. The evolution is initiated by numerical noise, undergoes the linear phase of exponen-tial growth, and saturates in the nonlinear regime. Within the linear phase the pulsation period and the growth rate can be compared with

the results of an independent linear stability analysis for validation.

A crucial test for the nonlinear simulation code is its behaviour in the initial phase of the evolution of a physical instability of an unstable model. Starting from an unsta-ble model in hydrostatic equilibrium as an initial condition the code should pick up one or more unstable modes with correct periods and growth rates (as predetermined by an independent linear stability analysis) from numerical noise without any further external perturbation. In particular, it should exhibit the linear phase of exponential growth of the instabilities, where the periods and growth rates predicted by the linear theory can be compared with the results of the simulations which is used to validate

-0.2 -0.1 0 0.1 0.2

0 15 30 45 60 75 90

Velocity[10−3cm/s]

Time (days)

(a) 10 M

-0.06 -0.03 0 0.03 0.06

0 10 20 30 40 50

Velocity[10−3cm/s]

Time (days)

(b) 20 M

FIGURE2.15: Photospheric velocity as a function of time for two stable ZAMS models with a mass of 10 M (a) and 20 M (b), respectively.

The models remain in hydrostatic equilibrium and the velocities shown correspond to numerical noise.

the code. The succesfull test proves that the simulated evolution is governed by physi-cal instabilities and not due any numeriphysi-cal instabilities or artifacts. As an example, Fig.

2.14shows the photospheric velocity as a function of time of an unstable ZAMS model with a mass of 90 Mas obtained by a simulation. The simulation starts in hydrostatic equilibrium with a velocity amplitude of the order of 10⁻⁵cm/s. After the linear phase of exponential growth it saturates in the nonlinear regime with an amplitude of ap-proximately 50 km/s after 16 days. An independent linear stability analysis provides the real (σr) and imaginary (σi) parts of the eigenfrequencies associated with the most unstable mode as:

• σ_r= 2.193 (corresponding to a period of 4.38 hours)

• σ_i= - 0.129

The growth rate determined from the simulation (see, Fig. 2.14) corresponds toσi= -0.127 and the period in the linear phase of exponential growth is found to be 4.3 hours ( corresponding to σ_r = 2.2). Both the period and the growth rate as obtained from the simulation match the period and the growth rate determined by the linear stability analysis. Hence the final result of the simulations in the nonlinear regime have to be regarded as the consequence of the physical instabilities of the model.

Simulations of unstable models have been demonstrated to pick the correct period and growth rate corresponding to the most unstable mode from numerical noise with-out any external perturbation. Vice versa, the simulation scheme might be tested by considering a stable hydrostatic model. In this case the model should remain in hy-drostatic equilibrium and the velocity amplitudes should stay on the numerical noise level. As an example, Fig. 2.15shows the velocity as a function of time as obtained by simulation for two stable hydrostatic ZAMS models with solar chemical compo-sition. As expected for a successful test, the photospheric velocity never exceeds the noise level of 10⁻⁴cm/s (see, Fig. 2.15). Thus the scheme is proven to be sensitive to physical instabilities and simultaneously does not suffer from numerical instabilities and artifacts.

Chapter 2. Basic equations and methods 29