High-resolution discretizations can be advantageous in compositional simulation to reduce excessive numerical diffusion that tends to mask shocks and fingering effects. In this work, we outline a fully implicit, dynamic, multilevel, high-resolution simulator for compositional problems on unstructured polyhedral grids. We rely on four ingredients: (i) sequential splitting of the full problem into a pressure and a transport problem, (ii) ordering of grid cells based on intercell fluxes to localize the nonlinear transport solves, (iii) higher-order discontinuous Galerkin (dG) spatial discretization with order adaptivity for the component transport, and (iv) a dynamic coarsening and refinement procedure. For purely cocurrent flow, and in the absence of capillary forces, the nonlinear transport system can be perturbed to a lower block-triangular form. With counter-current flow caused by gravity or capillary forces, the nonlinear system of discrete transport equations will contain larger blocks of mutually dependent cells on the diagonal. In either case, the transport subproblem can be solved efficiently cell-by-cell or block-by-block because of the natural localization in the dG scheme. In addition, we discuss how adaptive grid and order refinement can effectively improve accuracy. We demonstrate the applicability of the proposed solver through a number of examples, ranging from simple conceptual problems with PEBI grids in two dimensions, to realistic reservoir models in three dimensions. We compare our new solver to the standard upstream-mobility-weighting scheme and to a second-order WENO scheme.