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

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 10.4173/mic.2006.2.3. Experiments are run with implicit Runge-Kutta stages up to s=7, and it is found that the new preconditioner outperforms the others, with the improvement becoming more pronounced as spatial discretization is refined and as temporal order is increased.