Volume 10, Issue 2
A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids

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

Adv. Appl. Math. Mech., 10 (2018), pp. 261-274.

Published online: 2018-05

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  • 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 spatialmesh 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.

  • Keywords

Discontinuous Galerkin, Jacobian-free, implicit time-stepping, preconditioner.

  • AMS Subject Headings

65M10, 78A48

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-261, author = {}, title = {A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {2}, pages = {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 spatialmesh 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.}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0004}, url = {http://global-sci.org/intro/article_detail/aamm/12236.html} }
TY - JOUR T1 - A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 261 EP - 274 PY - 2018 DA - 2018/05 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2017-0004 UR - https://global-sci.org/intro/aamm/12236.html KW - Discontinuous Galerkin, Jacobian-free, implicit time-stepping, preconditioner. AB - 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 spatialmesh 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.
Wanglong Qin, Junwei Cai, Hongqiang Lu, Peter K. Jimack & M. A. Walkley. (2020). A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids. Advances in Applied Mathematics and Mechanics. 10 (2). 261-274. doi:10.4208/aamm.OA-2017-0004
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