Efficient Order-Optimal Preconditioners for Implicit Runge-Kutta and Runge-Kutta-Nystr\"om Methods Applicable to a Large Class of Parabolic and Hyperbolic PDEs

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.