TY - JOUR
T1 - A robust high-order residual distribution type scheme for steady Euler equations on unstructured grids
AU - Hu, Guanghui
AU - Li, Ruo
AU - TANG, Tao
N1 - Funding Information:
The research of Hu is supported by a studentship from Hong Kong Baptist University . The research of Li was supported in part by the National Basic Research Program of China under the Grant 2005CB321701 and the National Science Foundation of China under the Grant 10731060 . The research of Tang was supported in part by Hong Kong Research Grants Council , the FRG grants of Hong Kong Baptist University and an Cheung Kong Chair Professorship of the Chinese Ministry of Education through Beijing University of Aeronautics and Astronautics .
PY - 2010/3/1
Y1 - 2010/3/1
N2 - A robust high-order algorithm is proposed to solve steady Euler equations on unstructured grids. The main ingredients of the algorithm include a standard Newton method as the outer iterative scheme and a linear multigrid method as the inner iterative scheme with the block lower-upper symmetric Gauss-Seidel (LU-SGS) iteration as its smoother. The Jacobian matrix of the Newton-iteration is regularized by the local residual, instead of using the commonly adopted time-stepping relaxation technique based on the local CFL number. The local Jacobian matrix of the numerical fluxes are computed using the numerical differentiation, which can significantly simplify the implementations by comparing with the manually derived approximate derivatives. The approximate polynomial of solution on each cell is reconstructed by using the values on centroid of the cell, and limited by the WENO hierarchical limiting strategy proposed by Xu et al. [Z.L. Xu, Y.J. Liu, C.W. Shu, Hierarchical reconstruction for discontinuous galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, Journal of Computational Physics 228 (2009) 2194-2212]. It is found that the proposed algorithm is insensitive to the parameters used. More precisely, in our computations, only one set of the parameters (namely, the proportional constant α for the local residual, the relaxation parameter τ in the Newton-iteration, the weight μ in the WENO scheme and the number of smoothing steps in the multigrid solver) is employed for various geometrical configurations and free-stream configurations. The high-order and robustness of our algorithm are illustrated by considering two-dimensional airfoil problems with different geometrical configurations and different free-stream configurations.
AB - A robust high-order algorithm is proposed to solve steady Euler equations on unstructured grids. The main ingredients of the algorithm include a standard Newton method as the outer iterative scheme and a linear multigrid method as the inner iterative scheme with the block lower-upper symmetric Gauss-Seidel (LU-SGS) iteration as its smoother. The Jacobian matrix of the Newton-iteration is regularized by the local residual, instead of using the commonly adopted time-stepping relaxation technique based on the local CFL number. The local Jacobian matrix of the numerical fluxes are computed using the numerical differentiation, which can significantly simplify the implementations by comparing with the manually derived approximate derivatives. The approximate polynomial of solution on each cell is reconstructed by using the values on centroid of the cell, and limited by the WENO hierarchical limiting strategy proposed by Xu et al. [Z.L. Xu, Y.J. Liu, C.W. Shu, Hierarchical reconstruction for discontinuous galerkin methods on unstructured grids with a WENO-type linear reconstruction and partial neighboring cells, Journal of Computational Physics 228 (2009) 2194-2212]. It is found that the proposed algorithm is insensitive to the parameters used. More precisely, in our computations, only one set of the parameters (namely, the proportional constant α for the local residual, the relaxation parameter τ in the Newton-iteration, the weight μ in the WENO scheme and the number of smoothing steps in the multigrid solver) is employed for various geometrical configurations and free-stream configurations. The high-order and robustness of our algorithm are illustrated by considering two-dimensional airfoil problems with different geometrical configurations and different free-stream configurations.
KW - Block LU-SGS
KW - Hierarchical reconstruction
KW - High-order accuracy
KW - Multigrid
KW - Steady Euler equations
KW - WENO
UR - http://www.scopus.com/inward/record.url?scp=72449185727&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2009.11.002
DO - 10.1016/j.jcp.2009.11.002
M3 - Journal article
AN - SCOPUS:72449185727
SN - 0021-9991
VL - 229
SP - 1681
EP - 1697
JO - Journal of Computational Physics
JF - Journal of Computational Physics
IS - 5
ER -