  In this paper, we present a novel class of high-order Runge--Kutta (RK)
discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new
method extends beyond the traditional method of lines framework and utilizes
stage-dependent polynomial spaces for the spatial discretization operators. To
be more specific, two different DG operators, associated with $\mathcal{P}^k$
and $\mathcal{P}^{k-1}$ piecewise polynomial spaces, are used at different RK
stages. The resulting method is referred to as the sdRKDG method. It features
fewer floating-point operations and may achieve larger time step sizes. For
problems without sonic points, we observe optimal convergence for all the
sdRKDG schemes; and for problems with sonic points, we observe that a subset of
the sdRKDG schemes remains optimal. We have also conducted von Neumann analysis
for the stability and error of the sdRKDG schemes for the linear advection
equation in one dimension. Numerical tests, for problems including
two-dimensional Euler equations for gas dynamics, are provided to demonstrate
the performance of the new method.
