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

TL;DR

This paper introduces a novel block preconditioner based on LDU factorization with algebraic multigrid for large systems in implicit Runge-Kutta methods applied to parabolic PDEs, demonstrating superior performance in numerical tests.

Contribution

The paper presents a new block preconditioner that improves scalability and efficiency for implicit Runge-Kutta methods solving parabolic PDEs, outperforming existing methods.

Findings

Outperforms existing preconditioners in condition number and eigenvalue distribution.

More effective as spatial discretization is refined.

Provides better convergence with higher temporal order.

Abstract

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.