Title: A New Block Preconditioner for Implicit Runge-Kutta Methods for Parabolic PDE Problems

Abstract: Explicit time integrators for parabolic PDE are subject to a restrictive timestep limit, so A-stable integrators are essential. It is well known that although there are no A-stable explicit linear multistep methods and implicit multistep methods cannot be A-stable beyond order two, there exist A-stable and L-stable implicit Runge-Kutta (IRK) methods at all orders. IRK methods offer an appealing combination of stability and high order; however, these methods are not widely used for PDE because they lead to large, strongly coupled linear systems. An s-stage IRK system has s-times as many degrees of freedom as the systems resulting from backward Euler or implicit trapezoidal rule discretization applied to the same equation set. In this talk, I will introduce a new block preconditioner for IRK methods, based on a block LDU factorization with algebraic multigrid subsolves for scalability. I will demonstrate the effectiveness of this preconditioner on two test problems, a 2-D heat equation and a model advection-difusion problem. I compare this preconditioner in condition number and eigenvalue distribution, and in numerical experiments with other preconditioners currently in the literature. Experiments are run using IRK methods with two to seven stages. We find that the new preconditioner outperforms the others, with the improvement becoming more pronounced as spatial discretization is refined and as temporal order is increased.