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

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2. Numerical method

In what follows, we consider only the Direct Eulerian formulation of the PPM scheme as implemented in the PROMETHEUS code (Fryxell et al. 1989). Most of the presented results are valid for and can efficiently be implemented in codes based on the Lagrangian with remap approach, too. Both versions of the PPM method belong to the class of shock-capturing methods which are characterized by high-order spatial and temporal accuracy. In such schemes flow discontinuities are treated as solutions to the hydrodynamic equations (Riemann problem) and are represented by sharp (1 to 2 zones wide) and oscillation free profiles of hydrodynamic variables. There is no need for adding large amounts of artificial viscosity for shocks to obtain correct post-shock states. PROMETHEUS uses a Strang-type directional splitting (Strang 1968) for simulation of multidimensional flows.

For given initial data and boundary conditions each hydrodynamic step of the PPM scheme begins with construction of the interpolants approximating the distribution of flow variables inside each grid zone. The initial parabolic profiles are subsequently modified according to local flow conditions: density profiles are made steeper near contact discontinuities and the distributions of all variables are somewhat flattened near shocks in order to reduce high-frequency post-shock oscillations. Afterwards, monotonicity constraints are imposed on the interpolation profiles to avoid unphysical solutions. The monotonized profiles are used to calculate initial data for the Riemann problem at each zone interface by averaging the monotonized parabolae over the domain of dependence of the zone interface. The left and right states at the interface obtained in this way define the input data for the nonlinear Riemann problem, which is solved iteratively. The solution of the Riemann problem provides average hydrodynamic state variables at the zone interface, which are used to compute the fluxes for the advection step, whereby the hydrodynamic variables are advanced to the new time level.

For simulations of mixing in supernova explosions the basic PPM algorithm was modified to allow for advection of several nuclear species. The hydrodynamic state vector was extended by adding mass fractions for each of the species as were the left and right states used as input for the Riemann problem. The effective states at zone interfaces are obtained by averaging the mass fraction profiles over a properly chosen domain of dependence for the zone interface. The interpolation step for the mass fractions also includes steepening and flattening procedures followed by a monotonization step. The mass fractions do not enter the Riemann problem. They are treated as passive scalars and are advected with the flow depending on the upwind state as determined by the average velocity obtained from the solution of the Riemann problem.

PROMETHEUS includes the handling of a general equation of state using the approach of Colella & Glaz (1985). Gravitational forces are included in the calculation of the effective states entering the Riemann problem and by a separate acceleration step at the end of each hydrodynamic sweep. Operator splitting is also used to couple nuclear burning and hydrodynamics. The nuclear reaction network is solved using a multidimensional Newton iteration (Müller 1986). A more detailed description of the implementation of PROMETHEUS can be found in Fryxell et al. (1989) and Müller (1998).

2.1. Multi-fluid advection

We consider the one-dimensional initial-boundary value problem

[EQUATION]

with boundary data [FORMULA] and [FORMULA], where F , U and G are the flux vector, the state vector and the vector of source terms, respectively. In case of the Euler equations for multi-fluid ideal hydrodynamics the state vector is

[EQUATION]

where [FORMULA] and [FORMULA] are the total gas density, the velocity, the specific total energy, the specific internal energy, the pressure and the mass fraction and the partial density of the n-th fluid, respectively. The closure relation for the above system is provided by the equation of state, [FORMULA], where X is the vector of mass fractions.

Let the number of different fluid phases be [FORMULA] and the mass fraction of the n-th fluid inside zone i be [FORMULA]. Then the following relation must hold for [FORMULA],

[EQUATION]

2.2. FMA method

As it has been observed first by Fryxell et al. (1989) using PROMETHEUS and by Larrouturou (1991), the nonlinear character of higher-order Godunov-type schemes is the primary reason for the violation of (1). During a simulation it is not guaranteed that in such schemes the sum of the mass fractions inside each zone remains equal to unity even if both the underlying advection scheme is conservative and the total mass of each fluid (summed over the whole grid) remains constant to within machine accuracy (see also Müller 1998).

This failure of high-order schemes can be understood when one realizes that interpolation profiles are constructed independently for each mass fraction. Thus their sum can take an arbitrary value. We notice that this problem has a predominantly local character: (i) it is most important in regions where changes in composition are substantial, and (ii) condition (1) can be violated for any subvolume of a zone (e.g. for the zone of dependence).

In practice, condition (1) is usually enforced by applying a simple renormalization of the mass fractions after each step. This procedure, however, not only lacks any formal justification but it also leads to large systematic errors in the mass fractions of the least abundant species. It also violates the conservative character of the scheme.

One possible solution to this problem is to modify the interpolation step in such a way that deviations of the sum of the mass fractions from unity will remain small inside each zone. According to the procedure proposed by Fryxell et al. (1989), which we will refer to as FMA, this can be obtained by calculating the sum of the mass fractions at the zone interface and to flatten the interpolation profiles for that zone totally, if

[EQUATION]

with some predefined threshold value [FORMULA]. Here, and in what follows, we take expressions involving [FORMULA] to mean [FORMULA] equal to either + or -, where + and - refer to the right and left interface of the i-th zone, respectively.

In their original calculations Fryxell et al. (1989) used PROMETHEUS supplemented with FMA (with [FORMULA], while a less restrictive value of [FORMULA] seems to be acceptable for most applications). Although FMA acts only locally, in the long term it affects large regions of the grid due to its high diffusivity which effectively reduces the spatial accuracy of advection of species to first order. Consequently, one has to expect a large amount of mixing of fluids especially near composition interfaces.

2.3. CMA method

The Consistent Multi-fluid Advection (CMA) method retains the high accuracy of the PPM advection scheme for mass fractions with constraint (1) being accurately fulfilled, while any excessive flattening of interpolation profiles (as seen in case of FMA) is avoided.

The advection step can be written as,

[EQUATION]

where [FORMULA] (with [FORMULA] is the numerical flux vector across the zone interface of zone i. [FORMULA] is the area of the zone interface, [FORMULA] is the zone volume and [FORMULA] the size of the time step.

Since the PPM scheme is conservative,

[EQUATION]

the following condition holds for each component of [FORMULA] separately:

[EQUATION]

We seek for a set of numerical fluxes, [FORMULA], which satisfy conditions (1) and (5) simultaneously. In general, the total mass flux at a zone interface, [FORMULA], computed from the sum of the mass fluxes of individual species is not necessarily equal to the total mass flux computed with the (total) mass density [FORMULA]:

[EQUATION]

This inconsistency is the reason why any higher-order Godunov-type advection scheme in which the interpolation step for the mass fractions (or partial densities) is not appropriately constrained will violate condition (1). The inconsistency can be avoided when scaling the original partial mass fluxes [FORMULA] in such a way that their sum is exactly equal to the total mass flux.

We request the modified partial mass fluxes of zone i, [FORMULA] (which depend on the modified mass fractions at the interface [FORMULA] to be consistent with the advection of the total mass, i.e.

[EQUATION]

Hence, we modify the original partial mass flux vector using the simple scaling operation

[EQUATION]

in such a way that it becomes consistent with the continuity equation:

[EQUATION]

Comparing (7) with (9) one obtains

[EQUATION]

i.e. the normalization constant, [FORMULA], can be written in terms of the (unknown) modified mass fractions, [FORMULA], as

[EQUATION]

Since the modified mass fractions are consistent with advection of the total mass by definition (7), their sum is equal to unity, i.e.

[EQUATION]

In practice it is not necessary to explicitly compute the modified partial mass fluxes (8), but instead one simply has to scale the original mass fractions [FORMULA] according to (10), i.e.

[EQUATION]

The CMA method does not require any modification of the interpolation step of the advection scheme. Furthermore, flux scaling does not destroy the conservative character of the scheme (4). Hence, scaling the average mass fractions (13) obtained from the solution of the local Riemann problem by [FORMULA] defined in Eq. (12) is sufficient to satisfy both (1) and (5).

In passing we note that instead of scaling the partial mass fluxes one could, in principle, appropriately adjust the total mass flux, too (which is equivalent to computing the total mass flux as the sum of the partial mass fluxes). Although numerical experiments (not presented here) demonstrate that this method produces results of comparable quality to the CMA method, we prefer to use the latter because it preserves the original role of the total gas density in the PPM method. Finally, one could discard the continuity equation for the total density and use instead the partial densities as the primary variables to calculate the total density whenever needed. We found this method to give results of overall very poor quality, and being much more diffusive near contact and composition discontinuities.

Modification (13) implemented in the original PPM advection allows us to abandon the FMA scheme, and to retain the high accuracy of the PPM method. We will refer to this algorithm as sCMA. There are, however, still some problems left to be solved. In what follows, we focus on the problem of numerical diffusion across composition interfaces, and in these cases when the nonlinearity of the advection scheme becomes too strong to preserve the monotonicity of the scheme. This will complete the description of all of the elements constituting the CMA method.

FMA uses a contact steepening mechanism which in PPM is used to limit diffusion across contact discontinuities only. Removing FMA and using the full capabilities of the PPM interpolation algorithm for mass fractions, however, revealed a severe problem. Overshooting occurred near composition interfaces for the most abundant species. This, in turn, caused the least abundant species to be evacuated out of the critical region. Clearly, in its original version the PPM contact discontinuity detection algorithm cannot be directly used to steepen distributions of mass fractions. Some additional criteria have to be used to identify composition interfaces which need to be steepened and some mechanism has to be devised which limits overshooting. We point out that in most cases the FMA method effectively prevents any steepening of mass fractions profiles, since once any of the fluid distributions has been steepened, it likely activated the FMA flattening procedure.

Since we do not expect the geometrical properties of composition interfaces to be substantially different from that of contact discontinuities, most of the original steepening algorithm can be safely used without modification for detecting large gradients in the fluid's composition. It is very unlikely to find some additional constraint similar to that originally proposed for detection of contact discontinuities (CW, Eq. 3.2) in order to make the scheme more selective. Therefore, we will only consider local properties of composition profiles. One possibility is to associate composition interfaces with rapid changes in fluid composition. For this purpose we define a steepness measure for mass fraction profiles

[EQUATION]

Additional steepening is applied if [FORMULA] is larger than a fixed value [FORMULA]. Since [FORMULA] has a similar meaning as the parameter [FORMULA] used in PPM for calculation of the flattening coefficients (CW, Eq. A8), we set [FORMULA]. To prevent the steepening of relatively small composition jumps the following criterion has to be satisfied, too:

[EQUATION]

We use [FORMULA]. Furthermore, we do not steepen composition profiles near extrema. If the zone is located next to an extremum for which the following criterion has to be fulfilled

[EQUATION]

the steepening coefficient is set to zero. Finally, we do not steepen abundance profiles inside contact discontinuities, because this gives rise to enhanced overshooting of the partial densities, [FORMULA]. After steepening we flatten and monotonize the mass fraction profiles in the same way as in PPM.

As a remedy for the overshooting near composition discontinuities, we introduce two additional modifications. Local extrema of the mass fraction distributions are identified by the criterion

[EQUATION]

If the criterion is fulfilled in a zone i, the interpolation profiles are flattened by an additional (constant) amount in the two zones next to it [FORMULA]:

[EQUATION]

We use [FORMULA].

The second modification is somewhat more complex. For each zone we calculate two sums

[EQUATION]

at both zone interfaces [FORMULA], which give the total positive [FORMULA] and negative [FORMULA] deviation between the values of the mass fractions at the respective zone interface [FORMULA] and the values of the zone averaged mass fractions [FORMULA].

We then correct the interface values of the mass fractions according to (14). However, instead of the constant flattening coefficient w, we use a variable coefficient [FORMULA], which depends on the positive [FORMULA] and negative [FORMULA] deviations defined in (15):

[EQUATION]

where

[EQUATION]

[EQUATION]

and

[EQUATION]

with [FORMULA].

The reasoning behind this procedure is based on the observation that any variation in the distribution of one fluid component should be compensated by an appropriate variation of the distribution of at least one other component. Here, we define two separate groups of fluid components which show deviations (from the zone average) of the same sign at the zone interface. Once the sums of the negative [FORMULA] and the positive [FORMULA] deviations are obtained, we try to limit their relative difference. Note that the FMA flattening criterion is based on the absolute value of the deviation of the sum [FORMULA] from unity.

In the CMA method the amount of additional flattening is not constant but smoothly increases with the relative difference between the two deviations. Maximum flattening is applied if the relative difference between the two deviations exceeds [FORMULA], while no additional flattening is introduced for [FORMULA]. Hence, when [FORMULA], the interpolation profiles for that group of species which shows the largest absolute deviation are modified using the same flattening coefficient for all species. This reduces the deviation and brings it closer to that of the other group of species.

In this way we introduce a finite amount of coupling between the interpolation profiles of two distinct groups of fluid components rather than trying to adjust the distributions of fluid components individually. Since the additional flattening procedure has a strictly dissipative character and relies on flattening of interpolation profiles, it is very unlikely that it causes unphysical solutions.

Finally, we mention that the just described procedure could also be used to guarantee consistency between the total mass flux and the sum of the partial mass fluxes. Hence, it could be applied both in the interpolation and the advection step.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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