# Stage-parallel fully implicit Runge-Kutta solvers for discontinuous Galerkin fluid simulations

## TL;DR

This paper introduces a parallel-in-time fully implicit Runge-Kutta solver for discontinuous Galerkin fluid simulations, demonstrating improved efficiency and accuracy over existing methods through novel linear system transformations and preconditioning techniques.

## Contribution

It develops a new parallel-in-time fully implicit Runge-Kutta method with effective linear system transformations and preconditioners for fluid simulations.

## Key findings

- Radau IIA methods outperform DIRK in accuracy and efficiency
- Parallel-in-time approach reduces computational time
- Transformations decrease linear system solution costs

## Abstract

In this paper, we develop new techniques for solving the large, coupled linear systems that arise from fully implicit Runge-Kutta methods. This method makes use of the iterative preconditioned GMRES algorithm for solving the linear systems, which has seen success for fluid flow problems and discontinuous Galerkin discretizations. By transforming the resulting linear system of equations, one can obtain a method which is much less computationally expensive than the untransformed formulation, and which compares competitively with other time-integration schemes, such as diagonally implicit Runge-Kutta (DIRK) methods. We develop and test several ILU-based preconditioners effective for these large systems. We additionally employ a parallel-in-time strategy to compute the Runge-Kutta stages simultaneously. Numerical experiments are performed on the Navier-Stokes equations using Euler vortex and 2D and 3D NACA airfoil test cases in serial and in parallel settings. The fully implicit Radau IIA Runge-Kutta methods compare favorably with equal-order DIRK methods in terms of accuracy, number of GMRES iterations, number of matrix-vector multiplications, and wall-clock time, for a wide range of time steps.