skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

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

Journal Article · · Journal of Computational Physics
ORCiD logo [1];  [2]
  1. Brown Univ., Providence, RI (United States)
  2. Univ. of California, Berkeley, CA (United States)
+ Show Author Affiliations

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. In conclusion, 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.

Research Organization:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Univ. of California, Oakland, CA (United States)
Sponsoring Organization:
USDOE
Grant/Contract Number:
AC02-05CH11231
OSTI ID:
1543560
Alternate ID(s):
OSTI ID: 1398120
Journal Information:
Journal of Computational Physics, Vol. 335, Issue C; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 32 works
Citation information provided by
Web of Science

References (18)

Diagonally Implicit Runge–Kutta Methods for Stiff O.D.E.’s journal December 1977
Linearly implicit Rosenbrock-type Runge–Kutta schemes applied to the Discontinuous Galerkin solution of compressible and incompressible unsteady flows journal September 2015
Implicit Time Integration Schemes for the Unsteady Compressible Navier–Stokes Equations: Laminar Flow journal June 2002
On the implementation of implicit Runge-Kutta methods journal September 1976
Fourth-Order Runge–Kutta Schemes for Fluid Mechanics Applications journal October 2005
Order Results for Implicit Runge–Kutta Methods Applied to Stiff Systems journal June 1985
Validation of a High-Order Large-Eddy Simulation Solver Using a Vertical-Axis Wind Turbine journal January 2016
Singly Diagonally Implicit Runge–Kutta Methods with an Explicit First Stage journal August 2004
An incomplete factorization technique for positive definite linear systems journal May 1980
Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows: DG-MEBDF UNSTEADY SOLUTION OF COMPRESSIBLE FLOWS journal August 2014
Up to sixth-order accurate A-stable implicit schemes applied to the Discontinuous Galerkin discretized Navier–Stokes equations journal November 2014
The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems journal January 2008
Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier–Stokes Equations journal January 2008
High-order CFD methods: current status and perspective: HIGH-ORDER CFD METHODS journal January 2013
A class ofA-stable methods journal September 1969
A special stability problem for linear multistep methods journal March 1963
A parallelizable preconditioner for the iterative solution of implicit Runge–Kutta-type methods journal November 1999
High-Order Discontinuous Galerkin Methods for CFD book March 2011

Cited By (6)

Evaluation of Fully Implicit Runge Kutta Schemes for Unsteady Flow Calculations journal July 2017
Analysis and Entropy Stability of the Line-Based Discontinuous Galerkin Method journal March 2019
Analysis and entropy stability of the line-based discontinuous Galerkin method text January 2018
Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations journal December 2019
Irksome: Automating Runge--Kutta time-stepping for finite element methods preprint January 2020
Kernel-based Active Subspaces with application to CFD parametric problems using Discontinuous Galerkin method text January 2020

Similar Records

Related Subjects

97 MATHEMATICS AND COMPUTING
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
Computer Science
Physics
Implicit Runge–Kutta
Discontinuous Galerkin
Preconditioned GMRES
Parallel-in-time