**6. Conclusion**

In this chapter, we discussed the need for implicit algorithms for resistive magnetohydrodynamics. We highlighted two broad classes of nonlinear methods: Newton-Krylov and nonlinear multigrid. We illustrated two Newton-Krylov approaches for MHD which are essentially very similar in the overall approach, but differed in the preconditioning strategies for expediting the iterative solution steps in the Krylov linear solver stage of the overall method. One preconditioning strategy is based on a "parabolization" approach while the other utilizes the local wave structure of the underlying hyperbolic waves in the MHD PDEs. The literature on the use of nonlinear multigrid for MHD is essentially sparse and therein we focused on a defect-correction approach coupled with a point-wise Gauss-Seidel smoother utilizing a first order upwind approach. Both approaches are valid and have their place, but it is clear that the nonlinear multigrid approach for MHD is still relatively new and could be further developed.
