An efficient numerical solution scheme based on a new mapped finite difference discretization and iterative strategies is developed for submicron semiconductor devices. As a representative model we consider a nonparabolic hydrodynamic system. The discretization is formulated in a mapped reference domain, and incorporates a transformed Scharfetter-Gummel treatment for the current density and energy flux. This permits the use of graded, nonuniform curvilinear grids in the physical domain of interest, which has advantages when gridding irregular domain shapes or grading meshes for steep solution profiles. The solution of the discrete system is carried out in a fully coupled, implicit form, and nonsymmetric gradient-type iterative strategies are investigated. Numerical results demonstrating the performance and reliability of the scheme are presented for test problems.