Ivo Dravins: Stage-parallel preconditioning for implicit Runge–Kutta methods

Abstract:

Fully implicit Runge–Kutta methods offer the possibility to use high-order-accurate time discretization with desirable stability properties. For general implicit Runge–Kutta methods all stages are coupled leading to a potentially costly and involved solution procedure which has been a major barrier to their widespread use.

We present a stage-parallel block preconditioner for the class of L-stable Radau IIA Runge–Kutta methods. The preconditioner exploits a property of the coefficient matrices to construct a block lower-triangular preconditioner. During the application of the preconditioner, a basis change can be applied to obtain a block-diagonal form, in this way allowing us to decouple the stages when solving for the blocks. In the linear case this basis change can be applied directly -- for non-linear equations further approximations are needed to achieve this decoupling.

For the linear case, we discuss the analysis of the preconditioned matrices which are non-symmetric and in tensor form. We give eigenvalue bounds for the two- and three-stage methods under symmetric positive definite assumptions and discuss what can be inferred about the general case.

We illustrate the performance by numerical examples, including also applications to non-linear problems. Finally the parallel behavior is demonstrated, comparing space-parallel but serial in stages against fully stage-parallel implementations on HPC platforms.