A new block preconditioner for implicit Runge-Kutta methods for parabolic PDE problems

Md Masud Rana, Victoria E. Howle, Katharine Long, Ashley Meek, William Milestone

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


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.

Original languageEnglish
Pages (from-to)S475-S495
JournalSIAM Journal on Scientific Computing
StatePublished - 2020


  • Implicit Runge-Kutta
  • Parabolic PDE
  • Preconditioning


Dive into the research topics of 'A new block preconditioner for implicit Runge-Kutta methods for parabolic PDE problems'. Together they form a unique fingerprint.

Cite this