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

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 disconti...

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Bibliographic Details
Published in:Journal of computational physics 2017-04, Vol.335 (C), p.700-717
Main Authors: Pazner, Will, Persson, Per-Olof
Format: Article
Language:English
Subjects:
Aerodynamics
Algorithms
Computational fluid dynamics
Computational physics
Computer simulation
Discontinuous Galerkin
Eulers equations
Fluid dynamics
Fluid flow
Galerkin method
Implicit Runge–Kutta
Iterative methods
Linear systems
Mathematical analysis
Matrix algebra
Matrix methods
Navier-Stokes equations
Parallel-in-time
Preconditioned GMRES
Runge-Kutta method
Simulation
Solvers
Studies
System effectiveness
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2017.01.050