  A new preconditioner based on a block $LDU$ factorization 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 [{\em Modeling,
Identification and Control}, 27 (2006), pp. 109-123]. Experiments are run on
two test problems, a $2D$ 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.
