An iterative semi-implicit scheme for KAW-mediated magnetic reconnection

Recent results in the fields of magnetic reconnection and of the tearing mode instability have come to emphasize the importance of going beyond the simple, single fluid, MHD description. In particular, the Hall term and/or finite Larmor radius (FLR) effects have shown to be crucial in obtaining the long sought speed-ups of the reconnection rate (Birn et al., JGR 01; Aydemir, Phys Fluids B 92).

From the numerical point of view, these effects add an extra complication to the already difficult task of simulating the reconnection phenomenon: they introduce dispersive waves into the system, the whistler in the case of the Hall term, the kinetic Alfvén wave (KAW) in the case of the FLR effects. These waves have dispersion relations in which the frequency ~ 2 ? ? k , i.e., extremely fast when compared to the macroscopic dynamics of the system. Explicit integration schemes show great difficulty in coping with these waves, as the CFL condition scales like ?t ~ 1/? . This yields explicit timesteps which are extremely small and render impractical the explicit integration approach.

In this talk, we will discuss how the semi-implicit methods (Schnack et al., JCP 87) can be adapted to deal with the KAW. The main idea resides in deriving a wave-like operator which mimics the real wave operator in the linear and nonlinear regimes, while being analytically invertible, thus removing the necessity of purely implicit methods to invert the very large matrices that often arise. Timestep enhacements by factors of ~100 are obtained, with computational time per timestep roughly the same as with an explicit scheme. An error control method is derived and used to determine the timestep. This approach is thus both unconditionally stable and accurate. Comparisons with a purely explicit integration are shown and found to be in excellent agreement.