Hydromagnetic turbulence in computer simulations

Hydromagnetic processes play an important role in many astrophysical systems (e.g. stars, galaxies, accretion discs). This is because the medium is hot enough to be partially or fully ionized. Because of the huge scales involved the medium is usually turbulent, provided there is an instability (shear, convection) facilitating the cascading of energy down to small scales. In turbulence research it has been a long standing tradition to solve the hydrodynamic equations using spectral schemes which have the lowest possible discretization error. Spectral schemes are particularly useful for incompressible problems where one needs to solve a Poisson-type equation for the pressure. However, spectral schemes are no longer optimal in many astrophysical circumstances where flows are generally compressible. Lower order spatial derivative schemes are generally unacceptable in view of their low overall accuracy, even when schemes are used where mass, momentum, and energy are conserved to machine accuracy. On massively parallel machines, on the other hand, spectral schemes are difficult to make run efficiently. High order finite difference schemes are therefore a useful compromise. Such schemes can yield almost spectral-like accuracy. Our code uses centered finite differences which make the adaptation to other problems simple. Since the code is not written in conservative form, conservation of mass, energy and momentum can be used to monitor the quality of the solution. A third order Runge-Kutta scheme with 2N storage [1] is used for calculating the time advance.