Full Time Step Research Articles

Methods used in the past in order to solve numerically the resistive and compressible magnetohydrodynamic equations as an initial value problem are the two-step Lax-Wendroff method and the spectral method. Furthermore, it has recently been proposed to use direct finite space differences with a high-order time advance. This method should have considerably less numerical damping at small wavelengths than the Lax-Wendroff method. We will compare these three methods by solving the problem of reconnection in a periodic double current sheet system with superimposed magnetic field fluctuations. For the time advance we use the third-order Adams-Bashforth method. In addition to these methods we will study the behaviour of a fourth method based on the third-order Adams-Bashforth time advance and a determination of the spatial derivatives by cubic spline functions whose derivatives up to second order are continous at the grid points. In order to evaluate the various methods at small scalelengths, we analyze the time behaviour of the enstrophy (mean square vorticity) and of the maximum current density in the system. We found that the spectral method with wavenumbers |k| ≤32 and the spline method with a 64 × 64 grid give comparable results. The Lax-Wendroff method with a 128 × 128 grid (which contains the values at the half and at the full time step) gives results similar to the 64 × 64 finite difference method. However, both these methods have a large numerical dispersion and lead to unreasonably large values of the enstrophy and maximum current density. The finite difference method, although having less damping, has similar numerical dispersion properties as the Lax-Wendroff method. We then used the spline method in order to investigate the dependence of the reconnection process on the initial noise level. The maximum enstrophy produced during the magnetic island generation is directly proportional to the energy in the initial fluctuations.

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