  The Runge--Kutta (RK) discontinuous Galerkin (DG) method is a mainstream
numerical algorithm for solving hyperbolic equations. In this paper, we use the
linear advection equation in one and two dimensions as a model problem to prove
the following results: For an arbitrarily high-order RKDG scheme in Butcher
form, as long as we use the $P^k$ approximation in the final stage, even if we
drop the $k$th-order polynomial modes and use the $P^{k-1}$ approximation for
the DG operators at all inner RK stages, the resulting numerical method still
maintains the same type of stability and convergence rate as those of the
original RKDG method. Numerical examples are provided to validate the analysis.
The numerical method analyzed in this paper is a special case of the Class A
RKDG method with stage-dependent polynomial spaces proposed in
arXiv:2402.15150. Our analysis provides theoretical justifications for
employing cost-effective and low-order spatial discretization at specific RK
stages for developing more efficient DG schemes without affecting stability and
accuracy of the original method.
