In this two-part talk, we will present some recent efforts in improving the computational efficiency in simulating wave and kinetic transport models. In Part I, we will review the numerical stiffness of the standard discontinuous Galerkin (DG) methods of very high order accuracy, and the resulting stringent requirement for time step size in explicitly solving time dependent PDEs. We then report our work in overcoming such stiffness by devising energy-stable staggered DG methods for linear wave equations. In Part II, we consider the radiative transfer equation, a fundamental kinetic description of energy or particle transport through media involving scattering and absorption processes. One prominent computational challenge comes from the high dimensionality of the phase space. By leveraging the existence of a low-rank structure in the solution manifold induced by the angular variable in the scattering dominating regime, we propose and test a new reduced basis method.