  We generalize previous work by Mardal, Nilssen, and Staff (2007, SIAM J. Sci.
Comp. v. 29, pp. 361-375) and Rana, Howle, Long, Meek, and Milestone (2021,
SIAM J. Sci. Comp. v. 43, p. 475-495) on order-optimal preconditioners for
parabolic PDEs to a larger class of differential equations and methods. The
problems considered are those of the forms $u_{t}=-\mathcal{K}u+g$ and
$u_{tt}=-\mathcal{{K}}u+g$, where the operator $\mathcal{{K}}$ is defined by
$\mathcal{{K}}u:=-\nabla\cdot\left(\alpha\nabla u\right)+\beta u$ and the
functions $\alpha$ and $\beta$ are restricted so that $\alpha>0$, and
$\beta\ge0$. The methods considered are A-stable implicit Runge--Kutta methods
for the parabolic equation and implicit Runge--Kutta--Nystr\"om methods for the
hyperbolic equation. We prove the order optimality of a class of block
preconditioners for the stage equation system arising from these problems, and
furthermore we show that the LD and DU preconditioners of Rana et al. are in
this class. We carry out numerical experiments on several test problems in this
class -- the 2D diffusion equation, Pennes bioheat equation, the wave equation,
and the Klein--Gordon equation, with both constant and variable coefficients.
Our experiments show that these preconditioners, particularly the LD
preconditioner, are successful at reducing the condition number of the systems
as well as improving the convergence rate and solve time for GMRES applied to
the stage equations.
