**6.1 Future challenges**

In this chapter, we have focused exclusively on methods for single fluid resistive MHD. Future challenges will lie in the area of implicit methods for more complicated extended MHD models with FLR effects, several of which exhibit dispersive wave phenomena such as Whistler, Kinetic Alfvén waves, and gyroviscous waves. These dispersive high frequency waves essentially make the stable explicit time step proportional to the square of the mesh spacing, i.e., Δ*t* ∝ Δ*x*2; and hence the benefit from implicit methods is much more than those for single fluid MHD. Some progress in using Newton Krylov approaches for Hall-MHD has been reported by Chacón & Knoll (2003). However more work is required for general geometry, and inclusion of all dispersive wave families. Research in the area of nonlinear multigrid is essentially unexplored for extended MHD. Another interesting challenge in developing implicit methods for MHD is the combination of JFNK or FAS methods with adaptive mesh refinement (AMR). Some progress towards JFNK with AMR has been reported by Philip et al. (2008) on reduced incompressible MHD in 2D. Combining implicit methods with AMR will help mitigate not only the temporal stiffness issues but also help effectively resolve the range of spatial scales in MHD.
