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

A new preconditioner based on a block factorization into lower triangular, diagonal, and upper triangular factors (an $LDU$ factoriaztion) with algebraic multigrid subsolves for scalability is introduced for the large, structured systems appearing in implicit Runge--Kutta time integration of parabolic partial differential equations. This preconditioner is compared in condition number and eigenvalue distribution, and in numerical experiments with others in the literature: block Jacobi, block Gauss--Seidel, and the optimized block Gauss--Seidel method of Staff, Mardal, and Nilssen [ Model. Identif. Control, 27 (2006), pp. 109--123]. Experiments are run on two test problems, a two-dimensional heat equation and a model advection-diffusion problem, using implicit Runge--Kutta 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.