5. Numerical simulations of MHD equations
Many numerical schemes have been originally developed for hydrodynamics. Time-dependent MHD simulations are particularly challenging because of the wide range of time scales present in the MHD model. The extension of the hydrodynamic schemes to the magnetohydrodynamic equations is not straightforward. There are two principal difficulties associated with the numerical solution of the MHD equations as compared to the hydrodynamic equations (Roe & Balsara 1996). The first difficulty is that the MHD equations contain the magnetic field which must satisfy the divergence-free constraint, . Accumulation of the numerical errors associated with evolving the magnetic field components can lead to violation of this constraint, and eventually can force termination of the simulations. The second difficulty is that the MHD equations possess new families of waves and also admit a variety of exotic wave structures such as switch-on fast shocks, switch-off fast rarefactions, switch-off slow shocks, and switch-on slow rarefactions. It is also possible to obtain compound modes of either fast or slow waves. Hence, an MHD numerical scheme should accurately capture the entire range of such structures, which has a considerable impact on the performance of the algorithms which are required to provide the stable and accurate propagation of all new families of waves (e. g., Barmin et al. 1996). Roe & Balsara (1996) list the six cases that can potentially cause trouble.
Due to the intrinsic complexity of the MHD equations, the development of numerical techniques to solve them has been slower than for hydrodynamics. One of the special concerns when developing a scheme to solve MHD equations is that these equations are neither strictly hyperbolic nor strictly convex (e. g., Brio & Wu 1988). In practice this means that the wave speeds of two families may coincide, and that compound wave structures involving both shocks and rarefactions may sometimes develop.
The numerical algorithm used to solve the system of Eqs. (1)-(4) has been detailed elsewhere (DeVore 1991). Briefly, these equations are updated using COARC2D, a two-dimensional cold MHD code using a finite-volume method. Finite-volume methods are one of several different techniques available to solve the MHD equations. They are simple to implement, easily adaptable to complex geometries, and well suited to handle nonlinear terms. The code utilises the flux corrected transport method (Boris & Book 1973; Zalesak 1979; DeVore 1991; Murawski & Goossens 1994), which yields accurate results near steep gradients and moving contact discontinuities by adding numerical diffusion and antidiffusion fluxes to the scheme. The numerical algorithm consists of three stages: transport, diffusion, and anti-diffusion. As explained above, the transport stage introduces ripples into the solution. In the diffusion stage a strong numerical diffusion is added to the transported solution to remove the ripples from it. This numerical diffusion is subsequently compensated by introducing anti-diffusive fluxes, which should eliminate the ripples associated with dispersion and with the Gibbs phenomenon. The latter are `corrected' to preserve the positivity and monotonicity of the profiles. Therefore, at this stage the anti-diffusive fluxes are multiplied by a factor which should not create new or accentuate old extrema.
The resulting computer code (DeVore 1991) is fourth-order accurate in space and second-order accurate in time, and the divergence of the magnetic field is preserved to within machine-dependent round-off errors by placing the magnetic field components at the interface locations of the finite-difference grid. An explicit predictor-corrector method has been applied for discretisation in time. Consequently, the time step is effectively limited by the Courant-Friedrichs-Levy condition. The stability condition of the flux-corrected transport scheme requires . Somehow modified versions of this code were successfully applied to simulate the resonant absorption of nonlinear Alfvén waves (Murawski et al. 1996a) and in recent numerical studies of the solar wind interaction with cometary plasma (Murawski et al. 1996b).
The flux corrected algorithm belongs to the so called modern shock-capturing schemes (e. g. Woodward & Colella 1984). These schemes are intelligent as they add only enough dissipation in small localised regions to eliminate numerical oscillations. To achieve this the dissipation has to be nonlinear and implemented locally, at regions of high-gradients. Consequently, the schemes reduce to some highly dissipative, first-order schemes when the solutions are discontinuous and to some minimally dissipative, higher-order scheme when the solution is smooth. Dissipation is added only for those wavelengths that the high-order scheme cannot resolve. As a consequence of that, the shock is captured over a small number of points and without any oscillations.
© European Southern Observatory (ESO) 1998
Online publication: January 16, 1998