Avoiding Magnetic Monopoles in Numerical MHD Calculations

Abstract

Beside the equations for mass, momentum and energy conservation, a numerical magnetohydrodynamical (MHD) code (assuming infinite conductivity) has to solve the magnetic field equation:

$$\frac{\partial }{{\partial t}}\overrightarrow B = [\overrightarrow \triangleright \times [\overrightarrow v \times \overrightarrow B ]],$$

(1)

which automatically fulfills $(\overrightarrow \triangleright \cdot \overrightarrow B ) =0$ if used as an initial condition. However, due to discretization errors usually artificial magnetic monopoles develop and grow during the computation, what causes an artificial force parallel to the field, energy and momentum no longer being conserved (BRACKBILL and BARNES [1]). Stationarity may not be achieved. In the present study we discuss the effects of the monopoles, and we get rid of them in treating $(\overrightarrow \triangleright \cdot \overrightarrow B ) = 0$ as a dynamical condition: RAMSHAW [2] shows that field equation (1) and solenoidal condition together are equivalent with the two equations (for $(\overrightarrow \triangleright \cdot \overrightarrow B ) = 0$)

$$\frac{\partial }{{\partial t}}\overrightarrow B = [\overrightarrow \triangleright \times [\overrightarrow v \times \overrightarrow B ]] + \overrightarrow \triangleright \Psi (\overrightarrow x ,t),$$

(2)

$${\overrightarrow \triangleright ^2}\Psi = \frac{\partial }{{\partial t}}\left( {\overrightarrow \triangleright \cdot \overrightarrow B } \right).$$

(3)

To implement (2) and (3) to a MHD code, it is more convenient to define the potential (with (△t is the time step)

$$\Phi (\overrightarrow x ) = \int_{t - \triangleleft t}^t {\Psi (\overrightarrow x ,{t^/})d{t^/}} .$$

(4)

After having calculated $\overrightarrow B (\overrightarrow x ,t)$ from (1), we solve the Poisson equation

$${\overrightarrow \triangleright ^2}\Phi = - (\overrightarrow \triangleright \cdot \overrightarrow B ).$$

(5)

Finally $\overrightarrow B (\overrightarrow x ,t)$ from (1) has to be replaced by

$${\overrightarrow B ^/}(\overrightarrow x ,t) = \overrightarrow B (\overrightarrow x ,t) + \overrightarrow \triangleright \Phi ,$$

(6)

what is equivalent to solving (2) and (3), and yields the ordinary field equation (1) with $(\overrightarrow \triangleright \cdot \overrightarrow B )=0$ and $\tilde \Phi$=const.on the boundary surface.