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A new preconditioner based on a block $LDU$ factorization with algebraicmultigrid subsolves for scalability is introduced for the large, structuredsystems appearing in implicit Runge-Kutta time integration of parabolic partialdifferential equations. This preconditioner is compared in condition number andeigenvalue distribution, and in numerical experiments with others in theliterature: block Jacobi, block Gauss-Seidel, and the optimized blockGauss-Seidel method of Staff, Mardal, and Nilssen [{\em Modeling,Identification and Control}, 27 (2006), pp. 109-123]. Experiments are run ontwo test problems, a $2D$ heat equation and a model advection-diffusionproblem, using implicit Runge-Kutta methods with two to seven stages. We findthat the new preconditioner outperforms the others, with the improvementbecoming more pronounced as spatial discretization is refined and as temporalorder is increased.

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