A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids

A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids

Year:    2018

Author:    Wanglong Qin, Junwei Cai, Hongqiang Lu, Peter K. Jimack, M. A. Walkley

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 2 : pp. 261–274

Abstract

Storage requirement and computational efficiency have always been challenges for the efficient implementation of discontinuous Galerkin (DG) methods for real life applications. In this paper, a fully implicit Jacobian-Free Newton-Krylov (JFNK) method is developed in the context of DG discretizations for the three-dimensional compressible Euler and Navier-Stokes equations. Compared with the Jacobian-based methods, the Jacobian-Free approach saves the storage for the Jacobian matrix which can be of great importance for DG methods. Three types of preconditioners are investigated in which the block diagonal preconditioner requires the least storage, while the block LU-SGS and ILU0 preconditioners require more storage but are more computationally efficient. An implicit time-stepping strategy is adopted for the stability of the current solver, which is based upon a hexahedral spatial mesh and the nonlinear solver package Kinsol is used to improve the computational efficiency and robustness. Numerical results demonstrate that the preconditioned JFNK-DG solver can substantially reduce the storage requirement compared with the Jacobian based method without significantly compromising accuracy or efficiency. Furthermore, as a good compromise between efficiency and storage requirement, the ILU0 preconditioner shows the best choice of the preconditioners presented.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2017-0004

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 2 : pp. 261–274

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    14

Keywords:    Discontinuous Galerkin Jacobian-free implicit time-stepping preconditioner.

Author Details

Wanglong Qin

Junwei Cai

Hongqiang Lu

Peter K. Jimack

M. A. Walkley