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Astron. Astrophys. 327, 786-794 (1997)

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2. MHD Equations and strong form of conservation laws

The non-relativistic MHD equations for a perfectly conducting fluid are

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA], e and [FORMULA] are the fluid density, internal energy density and velocity, respectively; [FORMULA] is the magnetic flux density, and [FORMULA] is the acceleration due to gravity of the Sun. The factor [FORMULA] denotes the Lagrangian or convective derivative. The current density [FORMULA] is related to [FORMULA] by

[EQUATION]

The fluid equations are closed by an equation of state [FORMULA]

Eqs. (1)-(3) describe the conservation of mass, momentum and energy, and Eq. (4) is the induction equation which is a restatement of magnetic flux conservation. In order to minimize truncation errors associated with finite differencing, it is convenient to recast the above equations in strong conservation form, which may be derived by integrating Eqs. (1)-(4) over an arbitrary control volume and its surface. This control volume corresponds to an individual cell or zone, in the desired spatial grid.

Consider a moving finite control volume V(t) with surface S(t). Using the adaptive grid transport theorem (Winkler et al. 1984), the integration of Eqs. (1)-(3), yields

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] is the grid velocity measured with respect to the Eulerian frame. The factor [FORMULA] denotes the total derivative with respect to the moving frame

[EQUATION]

Similarly, integrating Eq. (4) over a moving surface element S(t) bounded by a moving circuit C(t), one derives the general form of Faraday's law (Jackson 1975),

[EQUATION]

The ZEUS-2D solves the MHD equations using a explicit, multistep (operator split) finite difference method, the details can be found in the papers of Stone & Norman (1992a, b). However, for the sake of completeness we briefly mention the salient features of the algorithm. The fluid Eqs. (6)-(8) are solved in two steps, called the source and transport steps. In the source step, the following equations are solved in finite difference form

[EQUATION]

[EQUATION]

where the volume integration is dropped and an artificial viscous stress tensor Q has been introduced to treat shock waves. Next, in the transport step, the advection of fluid and magnetic flux is treated by solving

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

We solve the finite-difference equations by expressing them in covariant form in a cylindrical coordinate system. We ignore the azimuthal coordinate since we assume axial symmetry. This reduces the number of independent variables to two, while all components of vectors and tensors are retained. This approach is sometimes referred to as MHD in 2.5D.

The constraint that the magnetic field remain divergence free is implemented in the numerical treatment using the constrained transport (CT) algorithm of Evans & Hawley (1988). This is achieved by using the integral formulation of the induction equation to evolve the magnetic flux (Eq. 16). In addition it is important to calculate the EMF ([FORMULA]) accurately and in a manner which ensures numerical stability. The manner in which this is implemented in the code is described in Stone & Norman (1992b).

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© European Southern Observatory (ESO) 1997

Online publication: April 6, 1998
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