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Astron. Astrophys. 337, 311-320 (1998)

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2. Governing equations and numerical procedure

2.1. Governing equations

The governing equations are the hydrodynamic equations of an isothermal gas with gravity. The selfgravity of the gas is neglected. These equations are written in integral form as,

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] represent density, velocity in x-y direction and pressure,[FORMULA] and ds are the differential volume and surface elements and ([FORMULA]) is the outward normal of the surface element. The pressure for an isothermal gas is defined by.

[EQUATION]

where c is the constant sound speed.

The two momentum equations can be also written in terms of the angular momentum and of the momentum in the local radial direction, leading to

[EQUATION]

[EQUATION]

[EQUATION]

These momentum equations are used in the angular momentum conserving (AMC) scheme which will be described later.

2.2. Initial and boundary conditions

Initially the whole computational field is filled with a uniform gas. The outer boundary condition is given by the analytic solution of Bisnovatyi-Kogan et al. (1979). An absorbing boundary condition, in which the density is very low and the flow velocity is zero, is used at the inner boundary.

2.3. Computational mesh

A two dimensional cylindrical grid is used for our computations. The mesh is divided uniformly in angular zones and is divided nonuniformly in radius. The mesh size in the radial direction ([FORMULA]) changes exponentially such as, [FORMULA].

The radius of the absorbing inner boundary ([FORMULA]) is 0.01 [FORMULA] and that of the outer boundary ([FORMULA]) is 10 [FORMULA]. The standard grid is made of 200 mesh points in the angular direction and 140 mesh points in the radial direction. For the fine grid, these numbers are 400 and 250, respectively. The finest mesh size in the radial direction ([FORMULA]) is 0.001 [FORMULA] in both the standard and the fine grid (see Table 1).


[TABLE]

Table 1. Computational grids used in the simulations


Under astrophysical conditions, the inner boundary is usually located at the magnetopause of the compact accreting object. This radius is in general much smaller than the inner boundary of our computational domain. But even if the two values are comparable one has to realize that the numerical boundary conditions do not describe the physical situation at the magnetopause which will be much more complex. The absorbing boundary condition at the inner radius is more or less an artificial construction. In order to study the influence of [FORMULA], computations with different values of [FORMULA] were carried out.

2.4. Finite volume method

By applying the hydrodynamic equations in integral form to a rectangular volume constructed by mesh lines, a finite volume formulation is obtained. In order to obtain high spatial resolution, a MUSCL type approach and van Albada's flux limiter are used. The numerical fluxes which are used by the MUSCL scheme are described in Sect. 2.5.

Hydrodynamic and gravitational terms are integrated simultaneously. A two-step Runge-Kutta method is used for time integration. This scheme is accurate to second order in both space and time.

2.5. Simplified flux splitting (SFS) for isothermal gas

The finite volume method for solving the hydrodynamic equations takes the approximation of the flux normal to the interface which are denoted by Eqs. ( 5),( 6) and ( 8).

The numerical flux in the MUSCL scheme is computed from the values defined on both sides of the cell interface by,

[EQUATION]

where [FORMULA] denote the values on the left(+) and right(-) side, and m is the mass flux. This flux is computed by solving the Riemann initial value problem using the left and right side physical values. It is in principle possible to solve this problem exactly, but this would require a very large amount of computer time. Therefore approximate but very fast algorithms have been developed.

The way in which these are calculated is vital for the MUSCL type scheme, since it influences the computer time, robustness and accuracy. Both the robustness for strong shock waves and the accuracy for slip surfaces are necessary for this study, because of the high mach number near the object and the strong shearing in the accretion column and the disk.

Many algorithms for the numerical flux have been developed. Approximate Riemann values of the fluxes were obtained by the Roe (1981) and (Chakravarthy & Osher 1982). Although they are exact for shocks or rarefactions of moderate strength, they give unphysical results for strong rarefactions or shocks respectively. Thus they are not robust enough for hypersonic computations like this study where very strong shocks and rarefactions are produced by the high speed flow accelerated by gravity. Flux vector splitting schemes (Steger & Warming 1981, van Leer 1982, Hänel & Schwane 1989) are simple and more robust, and it has been shown that they have enough accuracy for the shock tube problem or flows around airfoils. However they are not accurate for contact discontinuities such as slip surfaces, since they have excess numerical shear stress which influences the momentum transfer especially in accretion disks. Thus these existing schemes are not good enough for the supersonic accretion flow. Wada & Liou (1994) discussed these problems in the context of to aerospace applications. Note that we consider only discontinuities in one-dimension or those aligned to grid lines in multidimensional flows. In general, discontinuities do not align to grid lines and all schemes including exact Riemann solver can not capture them exactly. Nevertheless it has been recognized that those characteristics strongly affect the accuracy.

Recently Liou & Steffen (1993) developed an Advection Upstream Splitting Method (AUSM). Althogh the AUSM exhibits small overshoots at shocks, it is as simple and robust as flux vector splitting schemes and as accurate as the exact solver for contact discontinuities. Inspired by this work, a family of AUSM type schemes have been developed (Jounouchi et al. 1993, Wada & Liou 1994, Shima & Jounouchi 1997). It has been shown that these AUSM type schemes are sufficiently simple robust and accurate.

Jounouchi et al. (1993, see also Shima & Jounouchi 1997) developed a new AUSM type numerical flux scheme named Simplified Flux Splitting (SFS). They reduced the overshoot at the shock wave. Originally the SFS was developed for adiabatic gas, but it has been modified for an isothermal gas in this study. The SFS scheme for an isothermal gas can be written as follows.

[EQUATION]

where [FORMULA] is an average value of the pressure defined by the following relations:

[EQUATION]

[EQUATION]

[EQUATION]

and

[EQUATION]

The mass flux m is given by the flux vector splitting method as follows:

[EQUATION]

[EQUATION]

Note that the usage of the flux vector splitting does not degrade the accuracy of a contact discontinuity for isothermal gas, because at the slip surface the SFS scheme gives zero flux normal to the surface.

2.6. Angular momentum conserving scheme

Finite volume methods are usually based on the conservation laws of mass, linear momentum and total energy. Conservation of energy is not used in the present computations because the gas is isothermal. These conservation laws, of course, also imply the conservation of angular momentum for the exact solutions. The conservation of linear momentum, however, is just an approximation for the angular momentum in a discretized description. As the accuracy is at most second order in mesh size with any finite volume method, it depends on the numerical mesh.

If the accreting matter in the accretion column does not collide with itself and if it keeps some angular momentum , it will follow a Keplerian orbit and cannot accrete. Thus the formation of an accretion disk near the object directly reflects the angular momentum of the accreting matter, and it therefore depends crucially on the accuracy of the conservation of angular momentum.

The angular momentum conserving (AMC) scheme based on Eqs. (9) and (10) shows less mesh dependency than the linear momentum conserving (LMC) scheme as will be shown later. The AMC scheme can easily be obtained from the LMC scheme. In fact, the difference between the AMC and LMC computer codes is only a few lines.

2.7. Accurate local time stepping

The solutions of wind accretion are nonsteady in most cases. Therefore, the numerical scheme has to be accurate in time to capture the motion correctly. Two-step Runge-Kutta time stepping with a small Courant number (less than 0.2) was used for this purpose. Near the object, the flow has very high velocity and a fine mesh is required there for spatial resolution, thus, the size of the time step is very restricted by the Courant condition and very long computing times are required to reach physically meaningful solutions, when the usual uniform step is taken.

In order to avoid this problem, time accurate local time stepping was used in this investigation. First a global time step is defined, then the time step at each cell is divided into two if the Courant number is larger than our limit. This procedure is repeated until Courant condition is satisfied in all cells. The inner iteration is carried out in one global iteration when the local time step is smaller than the global time step The conservation laws are fulfilled by summing up the fractional fluxes correctly. A speed up by about a factor 20 was obtained by this procedure. However, over 100 hours of computer time on a 2 GFLOPS class supercomputer (Fujitsu VPP-300) were still needed for the fine mesh case.

2.8. Units

The obvious reference length is the accretion radius [FORMULA]. The fluid velocity far upstream, [FORMULA], is used as a reference velocity, thus the time unit is

[EQUATION]

The mass accretion rate is normalized by the two-dimensional Hoyle & Lyttleton rate given by

[EQUATION]

The angular momentum accretion rate is normalized by

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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