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Astron. Astrophys. 342, 300-310 (1999)

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3. Numerical procedure

The governing MHD equations described in the previous section are of the hyperbolic type and can be cast in the form

[EQUATION]

where U represents a conservation quantity, and F and G flux components in x and y directions, respectively. The explicit version of the well-known MacCormack scheme, which is second order accurate in both time and space (Hirsch 1990), is used to discretize the above equation. According to this method, the predictor step is calculated from

[EQUATION]

and the corrector step is given by

[EQUATION]

The final step is given as

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA], [FORMULA], subscripts [FORMULA] are grid indices, [FORMULA], [FORMULA], where [FORMULA] is the time step, and [FORMULA] and [FORMULA] are grid spacings in x and y directions, respectively. For simplicity, a uniform grid will be used in this work. The grid spacing is chosen such that the wave structure is adequately resolved and the numerical dissipation inherent in the scheme is minimized.

A numerical scheme can either be implicit or explicit. For implicit schemes, the time step does not have to be restricted by the CFL (Courant, Friedrichs, Lewy) condition (Hirsch 1990). Therefore, these schemes are valuable for problems where fast convergence to a steady state is desired. However, a special attention must be given to the numerical iteration process. The explicit scheme is straightforward in implementation, but its disadvantage is that the time step has to satisfy the CFL condition. The latter states that the grid spacing cannot exceed the distance covered by a disturbance, traveling with maximum characteristic speed within the numerical time step utilized in the simulations (Hirsch 1990). For wave problems, the advantage of the implicit scheme is also not obvious because the time step is limited by the wave structure. In general, at least 15 grid points within a wavelength are needed to resolve the wave structure; if the time steps used in the simulations are too large, then meaningless (unphysical) results can be obtained. Thus, in this paper, the explicit version of the MacCormack scheme is used. The time step [FORMULA] from the CFL condition is given by

[EQUATION]

where [FORMULA] is the maximum of all characteristic wave speeds. [FORMULA] is the Alfvén speed.

For problems which involve discontinuities in the physical variables, numerical viscosity is needed in the explicit MacCormack scheme to achieve numerical stability. Hence, a numerical diffusion term in the form of [FORMULA], [FORMULA] being the velocity or the magnetic field strength, will be added to the original set of governing equations; µ is called "artificial viscosity". The introduction of artificial viscosity gives rise to another time step [FORMULA]. To obtain numerical stability of the central discretization of the numerical diffusion term, the following criterion has to be satisfied for two-dimensional problems (Hirsch 1990):

[EQUATION]

The appropriate time step [FORMULA] then is the smaller one of the two time steps [FORMULA] and [FORMULA]:

[EQUATION]

Having described the numerical procedure, it is now required to specify boundary conditions for the finite computational domain used in these numerical simulations. For the wave propagation problem, open boundary conditions are desired. Different methods have been used to implement these boundary conditions (see Forbes & Priest 1987, and references therein). A commonly used method is known as the Sommerfeld radiation condition. Among the many versions of this method, the simple approach developed by Orlanski (1976) has been successfully applied to many wave problems. In this approach, the wave propagation speeds of the various physical quantities at the boundary points are calculated by using values of these quantities at the nearby interior grid points. There is no inward propagation of information from outside the computational domain. If the Sommerfeld radiation condition is given by

[EQUATION]

where U represents an arbitrary physical quantity and [FORMULA] the wave propagation speed, then the Orlanski prescription (note that the original Orlanski paper contains printing errors) is to compute

[EQUATION]

as well as

[EQUATION]

using a leap-frog finite difference scheme.

Another important issue in solving MHD equations is the numerical treatment of the solenoidal condition. It is well-known that an incorrect numerical treatment of the induction equation may lead to a non-solenoidal magnetic field that varies in time and introduces a non-physical force along the field lines. Several numerical treatments have been proposed (Brackbill & Barnes 1980; Marder 1987; Evans & Hawley 1988). In the latter paper, the authors utilized a numerical technique called CT (Constrained Transport) that allows transforming the induction equation in such a way that it always maintains vanishing divergence of the field components to within machine round-off error. The CT technique is also adopted in this paper.

Before presenting the results of our numerical simulations, it is necessary to describe the time-dependent perturbations that are introduced in the computational domain to generate the wave motions. In this paper, only sinusoidal velocity perturbations are considered: [FORMULA] [FORMULA], where V is a dimensionless quantity and can either be [FORMULA] or [FORMULA] and perc represents the ratio between the velocity amplitude, [FORMULA], and the reference sound speed, [FORMULA]. In addition, [FORMULA] and [FORMULA] represent the location of the velocity perturbation in the computational domain and will be specified for each considered case.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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