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

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.

28 pages, 11 figures

Cited by in corpus (9)

• On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows
• Analysis and entropy stability of the line-based discontinuous Galerkin method
• On the convergence of spectral deferred correction methods
• Kernel-based active subspaces with application to computational fluid dynamics parametric problems using the discontinuous Galerkin method
• High-order partitioned spectral deferred correction solvers for multiphysics problems
• Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow
• Optimal and Low-Memory Near-Optimal Preconditioning of Fully Implicit Runge-Kutta Schemes for Parabolic PDEs
• Improving Efficiency of Parallel Across the Method Spectral Deferred Corrections
• An hp Multigrid Approach for Tensor-Product Space-Time Finite Element Discretizations of the Stokes Equations