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We present an implementation of a fully stage-parallel preconditioner forRadau IIA type fully implicit Runge--Kutta methods, which approximates theinverse of $A_Q$ from the Butcher tableau by the lower triangular matrixresulting from an LU decomposition and diagonalizes the system with as manyblocks as stages. For the transformed system, we employ a block preconditionerwhere each block is distributed and solved by a subgroup of processes inparallel. For combination of partial results, we either use a communicationpattern resembling Cannon's algorithm or shared memory. A performance model anda large set of performance studies (including strong scaling runs with up to150k processes on 3k compute nodes) conducted for a time-dependent heatproblem, using matrix-free finite element methods, indicate that thestage-parallel implementation can reach higher throughputs when the blocksolvers operate at lower parallel efficiencies, which occurs near the scalinglimit. Achievable speedup increases linearly with number of stages and arebounded by the number of stages. Furthermore, we show that the presentedstage-parallel concepts are also applicable to the case that $A_Q$ is directlydiagonalized, which requires complex arithmetic or the solution of two-by-twoblocks and sequentializes parts of the algorithm. Alternatively to distributingstages and assigning them to distinct processes, we discuss the possibility ofbatching operations from different stages together.

Stage-parallel fully implicit Runge-Kutta implementations with optimal multilevel preconditioners at the scaling limit