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Astron. Astrophys. 342, 179-191 (1999)

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3. Results

We have performed several numerical tests to illustrate problems arising in multi-fluid flows (when simulated with PROMETHEUS ) and to demonstrate the capability of the CMA method.

3.1. Shock tube

The first problem is the shock tube test problem originally proposed by Sod (1978). We have modified this problem to include three passively advected fluids. The initial state for this problem (Fig. 1) is,




with a discontinuous distribution of [FORMULA],


and oscillating mass fractions of the two other fluids,


[FIGURE] Fig. 1. Initial distribution of fluid phases in Sod's shock tube problem: [FORMULA] - solid line; [FORMULA] - dashed line; [FORMULA] - dash-dotted line.

The simulations have been performed with an ideal gas equation of state with [FORMULA], and an equidistant grid of 100 zones. Reflecting boundary conditions were imposed on both ends [FORMULA] and [FORMULA] of the computational domain.

We obtained three sets of results for different code configurations: original PPM code (Fig. 2, top row), PPM with FMA (middle section, [FORMULA], and PPM with CMA (bottom part) but with no special modifications of the interpolation algorithm included (sCMA). The left panels of Fig. 2 show the distributions of the fluids ([FORMULA] - solid line, [FORMULA] - dashed, [FORMULA] - dash-dotted). The right panels give the deviations of the sum of the mass fractions from unity, [FORMULA] (see Eq. (2)), with the lower right plot in Fig. 2 illustrating the truncation error of the CMA method (Eqs. (12) and (13)).

[FIGURE] Fig. 2. Distributions of fluid phases (left ) and deviations of the sum of mass fractions from unity (right ) in Sod's shock tube problem at time [FORMULA] for three different runs: original PPM code (top ), PPM with FMA (middle ), and PPM with sCMA (bottom ). [FORMULA] - solid line; [FORMULA] - dashed line; [FORMULA] - dash-dotted line.

The comparison of mass fraction profiles of different models at [FORMULA] (Fig. 2) shows a good agreement for [FORMULA] except for some small amount of clipping near the extrema of [FORMULA] and [FORMULA]. However, we find large differences near the jump in [FORMULA] at [FORMULA]. The amplitude of the sinusoidal variation of [FORMULA] and [FORMULA] is significantly reduced in case of FMA, and the initial discontinuities in [FORMULA] are strongly smeared. The original PPM scheme and the sCMA scheme are much less diffusive. Both jumps in [FORMULA] are clearly separated and the profiles of [FORMULA] and [FORMULA] do smoothly vary near [FORMULA].

The differences are even larger near the right boundary of the computational domain (Fig. 2). PPM strongly violates condition (1). [FORMULA] deviates from unity by up to 6% (where [FORMULA] is discontinuous). The errors are much smaller for FMA the deviations only being of the order of [FORMULA] which is close to the chosen value of [FORMULA]. However, as already noted above, the smaller error is bought at the cost of a degraded resolution. The sCMA method gave the best result. It does not only advect the fluids with high accuracy, but it is also able to keep [FORMULA] at the level of machine accuracy. The only imperfectness one notices is some overshoot in the distribution of [FORMULA] just to the left of the larger discontinuity signaling the first sign of a need for additional modifications of the interpolation scheme of the mass fractions (especially near composition interfaces).

To observe the long term behaviour of the three schemes we continued our simulations up to [FORMULA] (more than 75 000 steps with a Courant number of 0.8). Fig. 3 shows the evolution of the maximum negative and positive deviations from condition (1) recorded for each time step together with the mean absolute value of the deviation from unity averaged over all zones. The results for the original PPM method (top panel in Fig. 3) indicate large variations (in excess of 20%) which would certainly destroy any solution sensitive to chemical composition. When using FMA we observe a rapid rise of the maximum error which levels off after slightly exceeding [FORMULA]. The results obtained with sCMA show a slow growth of the maximum and minimum deviations, which seem to saturate at later times.

[FIGURE] Fig. 3. Long-term behaviour of mean and extreme deviations from condition (1) in Sod's shock tube problem. Top: original PPM; middle: FMA; bottom: sCMA.

3.2. Two interacting blast waves

This test problem was originally proposed by Woodward (1982). It gained much of its popularity later when being used by Woodward & Colella (1984) in their study of various advection schemes in case of flows with strong shocks. In this problem the initial state consists of a low-pressure region located in the central part of the grid,


which is bounded by two regions of (different) high pressure




For our test runs we used three passively advected fluids with mass fractions that are initially varying smoothly across the entire grid (Fig. 4),


[FIGURE] Fig. 4. Initial distributions of fluid phases in the interacting blast waves test problem: [FORMULA] - solid line; [FORMULA] - dashed line; [FORMULA] - dash-dotted line.

Again, we use an ideal gas equation of state with [FORMULA]. The grid consists of 400 equidistant zones. Reflecting conditions are imposed at both grid boundaries.

The results of our simulations at [FORMULA] are shown in Fig. 5.

[FIGURE] Fig. 5. Distributions of fluid phases (left ) and deviations of the sum of mass fractions from unity (right ) in the interacting blast wave problem at time [FORMULA] for three different runs: original PPM code (top ), PPM with FMA (middle ), and PPM with sCMA (bottom ). [FORMULA] - solid line; [FORMULA] - dashed line; [FORMULA] - dash-dotted line.

Since this time the initial distributions of the fluids are smooth we do not expect to see any discontinuities in the distributions of the mass fractions at later times. However, a discontinuity is created in the [FORMULA] and [FORMULA] distributions at [FORMULA] when using the original PPM and the sCMA method (left column, top and bottom panel of Fig. 5, respectively). No such discontinuity is present in the FMA data (middle panel). We identify the creation of such spurious composition interfaces with the actual failure of the unmodified discontinuity detection procedure of PPM (see Sect. 2.3). FMA once again proves to be more diffusive than the other two schemes: high-amplitude variations of [FORMULA] and [FORMULA] as seen in the PPM and sCMA data for [FORMULA] have markedly smaller amplitudes when calculated with FMA. Moreover, there is no trace of an extremum in [FORMULA] at [FORMULA]. In other parts of the grid all three methods produce very similar results.

As in the case of Sod's shock tube problem the condition (1) is most strongly violated when the original PPM method is used (upper left panel in Fig. 5). The maximum deviation of 2% occurs in that region where the FMA results are mostly affected by the use of an additional flattening procedure. On the other hand, FMA violates condition (1) at the level of [FORMULA] with a single pronounced maximum at the spurious composition interface created in the other two schemes. The sCMA method produces the most accurate results both during the initial phases of the evolution and in the long term evolution (lower left panel in Fig. 6). The maximum deviations from (1) exceed 10% for the original PPM method and fluctuate between 2 and 3 times [FORMULA] in case of FMA.

[FIGURE] Fig. 6. Long-term behaviour of mean and extreme deviations from condition (1) in the colliding blast waves problem. Top: original PPM; middle: FMA; bottom: sCMA.

3.3. Shock-contact interaction

The initial state for this problem is,





with [FORMULA] for [FORMULA], [FORMULA] for [FORMULA], and


The initial abundances with a composition interface between [FORMULA] and [FORMULA] at [FORMULA] are shown in Fig. 7. Again an ideal gas equation of state with [FORMULA] is used. The grid contains 400 equidistant zones. The left grid boundary is reflecting, while a flow-in boundary condition is imposed at the right grid boundary. The state of the inflowing gas is equal to that of the gas located near that boundary at the initial time.

[FIGURE] Fig. 7. Initial distributions of fluid phases for the shock-contact interaction problem: [FORMULA] - solid line; [FORMULA] - dashed line; [FORMULA] - dash-dotted line.

The initial conditions create a strong shock wave at [FORMULA] which propagates towards the left, hits the composition interface (initially located at [FORMULA] and then collides with the strong contact discontinuity (initially located at [FORMULA] that slowly moves to the left. Upon interaction a pair of shocks is generated.

Fig. 8 shows the distribution of the mass fractions together with the deviations from the condition (1) at [FORMULA] after all strong interactions have already taken place. All three methods give comparable results in regions of pure advective flow [FORMULA]. Towards the left follows a region with low-amplitude variations in the distributions of [FORMULA] and [FORMULA], which is much more diffused when calculated with FMA (middle left panel in Fig. 8). The composition interface [FORMULA] also seems to be smeared out in case of FMA, but it remains sharp in the other two cases. Finally, there is some overshoot in the distribution of [FORMULA] which can be seen just to the right of the composition interface in the sCMA data. The interface is slightly broader when calculated with the sCMA method than with the original PPM method primarily because of the smooth rounded profile associated with the overshoot.

[FIGURE] Fig. 8. Distributions of fluid phases (left ) and deviations of the sum of mass fractions from unity (right ) in the shock-discontinuity interaction problem at time [FORMULA] for three different runs: original PPM code (top ), PPM with FMA (middle ), and PPM with sCMA (bottom ). [FORMULA] - solid line; [FORMULA] - dashed line; [FORMULA] - dash-dotted line.

The analysis of deviations from condition (1) (right column in Fig. 8) confirms our conclusions from the previous tests. Original PPM produces deviations of the order of a few percent. Most of the extra diffusion used by FMA to keep deviations small occurs in the region of interaction between the shock and the contact discontinuity, and near the composition interface. It is in this region where differences in abundances between FMA and the other two codes are most apparent. The long term behaviour of the deviations (Fig. 9) indicates that there is no tendency for deviations to become smaller with time. Using the original version of PPM we find errors in the several percent range (up to 10%). The typical deviations for FMA are between 2 and 3 times [FORMULA]. There is some systematic increase in deviations visible in case of sCMA at late times, but it does not seem to be of any importance.

[FIGURE] Fig. 9. Long-term behaviour of mean and extreme deviations from condition (1) in the shock-contact interaction problem. Top: original PPM; middle: FMA; bottom: sCMA.

3.4. Supernova explosion

For our final "numerical" example we have chosen the shock propagation during the post-bounce evolution of a core collapse supernova. We consider the very early [FORMULA] stages of the evolution, when the just born supernova shock begins to sweep through the stellar layers just outside the iron core triggering nuclear synthesis. We will focus our attention on the role which numerical diffusion plays in the process of nuclear burning and on its impact on the final chemical composition of the thermonuclear processed material.

For these calculations we have used PROMETHEUS in a version which allowed us to consider those physical processes which play an important role during the early phases of the shock propagation. The gas is assumed to be a mixture of 14 nuclei ([FORMULA]H , [FORMULA]He , [FORMULA]C , [FORMULA]O , [FORMULA]Ne , [FORMULA]Mg , [FORMULA]Si , [FORMULA]S , [FORMULA]Ar , [FORMULA]Ca , [FORMULA]Ti , [FORMULA]Cr , [FORMULA]Fe , [FORMULA]Ni). An [FORMULA]-network with 27 reactions coupling 13 of these nuclei (all except [FORMULA]H) is used to describe nuclear burning. The approximate equation of state includes contributions from radiation, the 14 fully ionized Boltzmann gases, and positron-electron pairs (included in an approximate way). Self-gravity of the stellar envelope is taken into account as well as the gravitational attraction of a central point source which mimics the nascent neutron star [FORMULA].

The simulations have been performed in one spatial dimension assuming spherical symmetry. At the inner edge of the grid a reflecting boundary condition is imposed, while free outflow is allowed through the outer boundary. The inner boundary is situated at [FORMULA]cm corresponding to a mass coordinate of [FORMULA], which is well inside the Si shell. The computational domain, which extends out to a radius of [FORMULA]cm, is covered by 1600 zones (corresponding to a resolution of 40 km) in our standard resolution run. Additional simulations with up to 12 800 zones (corresponding to a resolution of 5 km) have been performed to study the convergence behaviour of different advection schemes.

The initial model is a [FORMULA] pre-collapse model of a blue supergiant (S. Woosley, private communication), which closely resembles a progenitor model of SN 1987A (Woosley, Pinto & Ensman 1988). The density, velocity and temperature profiles of the initial model are shown in Fig. 10.

[FIGURE] Fig. 10. Density, velocity, temperature and composition profiles for the most abundant species (from top to bottom ) of the [FORMULA] progenitor model. The high temperature region in the innermost [FORMULA] part of the Si shell results from the deposition of [FORMULA]ergs of internal energy in order to initiate the explosion.

In order to launch the supernova shock wave, we created a thermal bomb by adding [FORMULA]ergs in form of internal energy to the innermost [FORMULA] parts of the Si shell. The resulting shock wave propagates rapidly outwards heating up the stellar matter. Simultaneously a much weaker reverse shock propagates inwards. The temperatures and densities behind the supernova shock are sufficiently high to trigger thermonuclear burning and the production of new elements. The resulting release of nuclear energy slightly enhances the explosion energy. Whenever the outward propagating shock crosses one of the composition interfaces of the progenitor star (see Fig. 10), a weak reflected shock is created. The reflected shocks move inwards reheating the matter, which has just been processed by the supernova shock (see velocity profile in Fig. 11). A strong contact discontinuity at [FORMULA]cm corresponding to the mass coordinate [FORMULA] separates the shocked envelope gas from matter initially belonging to the thermal bomb.

[FIGURE] Fig. 11. Density (top ), velocity (middle ) and temperature (bottom ) profiles of the progenitor at [FORMULA]s obtained with the CMA method and a grid resolution of [FORMULA]km. The supernova shock has reached the helium shell and hence has already passed several composition interfaces. At every interface a weak reflected shock is created, which can be recognized in the velocity profile. The large density jump near [FORMULA]cm separates shocked gas that initially formed the thermal bomb from matter in the stellar envelope.

Fig. 12 shows the evolution of the chemical composition obtained with the FMA method at a resolution of 40 km in a narrow mass range [FORMULA] close to the outer edge of the thermal bomb. Nuclear burning is most intense very early on [FORMULA] when matter is still dense and hot. At later times the chemical composition does not change appreciately with one exception. Production of [FORMULA]Ti only begins around [FORMULA]ms (a barely visible bump between 1.355 and [FORMULA] and lasts for [FORMULA]ms.

[FIGURE] Fig. 12. Chemical composition of the ejecta obtained with the FMA method as a function of mass coordinate at [FORMULA], 100, 300, and 500 ms (from top to bottom ). The grid resolution is 40 km.

In many aspects the evolution is similar when calculated with the CMA method at the same grid resolution (Fig. 13). However, some important differences also exist. With the CMA method mixing between [FORMULA]O and other nuclear species at [FORMULA] is greatly reduced. [FORMULA]Ar and [FORMULA]Ca are clearly separated from oxygen at [FORMULA]ms. There is an indication of a [FORMULA]Si interface near [FORMULA]. The transition region extends only over two zones (over about 5 zones in case of FMA) and is accompanied by a small amount of overshooting towards larger radii. This region is difficult to model due to the presence of the strong contact discontinuity separating the stellar envelope from the thermal bomb. Without additional flattening and monotonization of the mass fraction profiles (as described in Sect. 2.3) the overshooting of [FORMULA]Si is suspiciously large. We note that overshooting of the most abundant species near composition interfaces might be a common problem for hydrodynamic codes (see, for example, Fig. 4 of Woosley, Pinto & Ensman 1988), and certainly deserves further investigation.

[FIGURE] Fig. 13. Chemical composition of the ejecta obtained with the CMA method as a function of mass coordinate at [FORMULA], 100, 300, and 500 ms (from top to bottom ). The grid resolution is 40 km.

Another difference between the FMA and CMA results is the mass of [FORMULA]Ti produced in the simulations, which seems to be quite sensitive to the amount of numerical diffusion. Since titanium is synthesized via the reaction [FORMULA]Ca [FORMULA] [FORMULA]Ti and since enhanced diffusion results in a smoother distribution of calcium, we computed several models where the interpolation profile for calcium was constructed assuming four different constant flattening coefficients [FORMULA] (models [FORMULA], [FORMULA], [FORMULA] and [FORMULA], respectively). An additional extreme model (CMAZ) was calculated where all mass fraction profiles are flattened completely thereby imposing a maximum amount of numerical diffusion. The results (Fig. 14) show that the amount of [FORMULA]Ti is smallest when using CMA [FORMULA], that it increases linearly with [FORMULA] and that the result of FMA [FORMULA] is recovered when the calcium profile is kept totally flat throughout the simulation (model [FORMULA]. In case of maximum numerical diffusion (model CMAZ) the amount of titanium [FORMULA] is even larger and exceeds that obtained with CMA by a factor of more than three. Table 1 summarizes these results; data taken from model 15A of Woosley, Pinto & Ensman (1988) (who used a different mechanism to initiate the explosion!) are also shown for comparison.

[FIGURE] Fig. 14. Total mass of [FORMULA]Ca (upper part ) and [FORMULA]Ti (lower part ) as a function of time obtained with different advection schemes. Solid lines: CMAZ (thin), FMA (medium), CMA (thick). CMA results with additional flattening for [FORMULA]Ca are shown by dashed lines for [FORMULA] equal to 0.25 (thin), 0.50 (thick), 0.75 (long thin), and 1.00 (long thick). The scale on the right side gives the masses normalized to the respective final mass obtained with CMA.


Table 1. Total masses of [FORMULA]Ca and [FORMULA]Ti (in units of [FORMULA] at [FORMULA]s a Model 15A of Woosley, Pinto & Ensman (1988).

Fig. 15 shows the composition profiles in the ejecta at [FORMULA]s for our three basic models: CMAZ (top), FMA (middle), and CMA (bottom). In CMAZ all abundances change smoothly and no particular feature can be recognized. In CMA there exist composition interfaces of [FORMULA]Ca [FORMULA], [FORMULA]O and [FORMULA]S [FORMULA], and several discontinuities (in [FORMULA]Si , [FORMULA]S , [FORMULA]Ar , [FORMULA]Ca) at [FORMULA]. Out of these only a relatively weak discontinuity in the distribution of [FORMULA]Ar can be recognized in the FMA model.

[FIGURE] Fig. 15. Composition structure of the progenitor at [FORMULA]s: CMAZ (top ), FMA (middle ), CMA (bottom ).

Finally, we have studied the convergence properties with respect to the production of heavy elements in our three schemes. The results for [FORMULA]Ti (Fig. 16) indicate that with CMAZ and FMA the production of titanium decreases as the resolution is improved. More interestingly, the CMA results depend only weakly on the resolution. This behaviour can be understood if we realize that there exists a composition interface in calcium near which titanium is formed, and that (as demonstrated by our numerical experiments; Fig. 14) the production of titanium grows with the diffusivity of the advection scheme. Once the composition interface is resolved and properly handled by the code, the final mass of titanium becomes practically independent of the spatial grid resolution.

[FIGURE] Fig. 16. Dependence of the production of [FORMULA]Ti on the grid resolution. The total [FORMULA]Ti mass is shown as a function of resolution at [FORMULA]s for models CMAZ (open squares), FMA (open circles), and CMA (full circles), respectively.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998