基于高阶耗散紧致格式的GMRES方法收敛特性研究
收稿日期: 2013-06-25
修回日期: 2013-10-12
网络出版日期: 2013-11-16
基金资助
国家自然科学基金（11072259）；国家“973”计划（2009CB723801）
Convergence Property Investigation of GMRES Method Based on High-order Dissipative Compact Scheme
Received date: 2013-06-25
Revised date: 2013-10-12
Online published: 2013-11-16
Supported by
National Natural Science Foundation of China (11072259); National Basic Research Program of China (2009CB723801)
计算效率较低是当前限制高阶精度计算方法应用的重要因素。为了提高高阶精度混合型耗散紧致格式（HDCS）的计算效率，发展了适合多块对接网格的广义最小残值（GMRES）方法，并利用GMRES方法开展了HDCS格式的加速收敛研究。首先研究了GMRES的预处理方法、CFL数和内层迭代步数对HDCS数值模拟收敛特性的影响，计算结果显示：点松弛方法是一种高效的预处理方法；CFL数对计算收敛速度影响较大；GMRES方法存在最优的内层迭代步数。利用GMRES方法完成了NACA 0012翼型绕流、NLR 7301翼型绕流和DLR-F4翼身组合体绕流的数值模拟，并与其他隐式时间推进方法进行了对比，GMRES方法计算更加稳定，并且计算效率相对LU-SGS（Lower-Upper Symmetric Gauss-Seidel）方法可以提高5倍以上。研究结果表明，本文发展的GMRES方法在多块对接网格中具有良好的计算稳定性，计算结果的残差可以收敛到更低的量级，并且可以较大幅度地提高高阶精度数值模拟的计算效率。
燕振国 , 刘化勇 , 毛枚良 , 邓小刚 , 朱华君 . 基于高阶耗散紧致格式的GMRES方法收敛特性研究[J]. 航空学报, 2014 , 35(5) : 1181 -1192 . DOI: 10.7527/S1000-6893.2013.0430
Low computational efficiency is an important factor constraining the application of high-order numerical methods. To improve the computational efficiency of hybrid cell-edge and cell-node dissipative compact scheme (HDCS), a generalized minimum residual (GMRES) algorithm suitable for multi-block structured grids is developed to accelerate simulations. The influence of GMRES's precondition methods, CFL number and sub-iteration number on convergence property of HDCS high-order simulations is investigated. It is shown that the point relaxation method is an efficient precondition method, that the CFL number can greatly affect the computational efficiency, and that GMRES has an optimal sub-iteration number. GMRES is applied to simulations of NACA 0012 airfoil, NLR 7301 airfoil and DLR-F4 wing/body configuration, and is compared with other implicit time integration methods. By using GMRES, the computation becomes more stable, and the computational efficiency can be improved by more than 5 times when compared with the LU-SGS(Lower-Upper Symmetric Gauss-Seidel) method. The results indicate that the GMRES method developed in this paper has good stability in multi-block structured grids, the residual can converge to lower levels, and GMRES can greatly improve the computational efficiency of high-order simulations.
[1] Jiang G S, Shu C W. Efficient implementation of weighted eno schemes[J]. Journal of Computational Physics, 1996, 126(1): 202-228.
[2] Deng X G, Zhang H X. Developing high-order weighted compact nonlinear schemes[J]. Journal of Computational Physics, 2000, 165(1): 22-44.
[3] Deng X G, Mao M L, Jiang Y, et al. New high-order hybrid cell-edge and cell-node weighted compact nonlinear schemes, AIAA-2011-3857.Reston: AIAA, 2011.
[4] Deng X G, Jiang Y, Mao M L, et al. Developing hybrid cell-edge and cell-node dissipative compact scheme for complex geometry flows//The Ninth Asian Computational Fluid Dynamics Conference, 2012: 1-11.
[5] Deng X G, Jiang Y, Mao M L, et al. Developing hybrid cell-edge and cell-node dissipative compact scheme for complex geometry flows[J]. Science China Technological Sciences, 2013, 56(10): 2361-2369.
[6] Zhang L P, Liu W, He L X, et al. A class of hybrid dg/fv methods for conservation laws I: basic formulation and one-dimensional systems[J]. Journal of Computational Physics, 2012, 231(4): 1081-1103.
[7] Deng X G, Mao M L, Tu G H, et al. Geometric conservation law and applications to high-order finite difference schemes with stationary grids[J]. Journal of Computational Physics, 2011, 230(4): 1100-1115.
[8] Deng X G, Min Y B, Mao M L, et al. Further studies on geometric conservation law and applications to high-order finite difference schemes with stationary grids[J]. Journal of Computational Physics, 2013, 239: 90-111.
[9] Jiang Y, Deng X G, Mao M L, et al. Large eddy simulation based on seventh-order dissipative compact scheme //The Ninth Asian Computational Fluid Dynamics Conference, 2012: 1-9.
[10] Jiang Y, Mao M L, Deng X G, et al. Effect of surface conservation law on large eddy simulation based on seventh-order dissipative compact scheme[J]. Applied Mechanics and Materials, 2013, 419: 30-37.
[11] Rizzetta D P, Visbal M R, Blaisdell G A. A time-implicit high-order compact differencing and filtering scheme for large-eddy simulation[J]. International Journal for Numerical Methods in Fluids, 2003, 42(6): 665-693.
[12] Ekaterinaris J A. Implicit, high-resolution, compact schemes for gas dynamics and aeroacoustics[J]. Journal of Computational Physics, 1999, 156(2): 272-299.
[13] Parsani M, Abeele K V D, Lacor C, et al. Implicit LU-SGS algorithm for high-order methods on unstructured grid with p-multigrid strategy for solving the steady navier-stokes equations[J]. Journal of Computational Physics, 2010, 229(3): 828-850.
[14] Zhang Y F. Investigations of convergence acceleration and complex flow numerical simulation for high-order accurate scheme (wcns). Mianyang: China Aerodynamics Research and Development Center, 2007.(in Chinese) 张毅锋. 高精度格式(WCNS)加速收敛和复杂流动数值模拟的应用研究. 绵阳: 中国空气动力研究与发展中心, 2007.
[15] Whitfield D L, Taylor L K. Discretized newton-relaxation solution of high resolution flux-difference split schemes, AIAA-1991-1539. Reston: AIAA, 1991.
[16] Buelow P E O, Venkateswaran S, Merkle C L. Stability and convergence analysis of implicit upwind schemes[J]. Computers & Fluids, 2001, 30(7-8): 961-988.
[17] Ezertas A A, Eyi S. Performances of numerical and analytical jacobians in flow and sensitivity analysis, AIAA-2009-4140.Reston: AIAA, 2009.
[18] Knoll D A, Keyes D E. Jacobian-free newton-krylov mehtods: a survey of approaches and applications[J]. Journal of Computational Physics, 2004, 193(2): 357-397.
[19] Cockburn B, Karniadakis G, Shu C W. Discontinuous galerkin methods: theory, computation and applications [M]. New York: Springer, 2000: 197-208.
[20] Persson P O, Peraire J. Newton-GMRES preconditioning for discontinuous galerkin discretizations of the Navier-Stokes equations[J]. SIAM Journal on Scientific Computing, 2008, 30(6): 2709-2733.
[21] Nejat A, Ollivier-Gooch C. Effect of discretization order on preconditioning and convergence of a high-order unstructured newton-gmres solver for the euler equations[J]. Journal of Computational Physics, 2008, 227(4): 2366-2386.
[22] Yoon S, Jameson A. Lower-upper symmetric gauss-seidel method for the euler and navier-stokes equations[J]. AIAA Journal, 1988, 26(9): 1025-1026.
[23] Yang X Q, Yang A M, Sun G. An efficient numerical method for coupling the rans equations with spalart-allmaras turbulence model equation[J]. Acta Aeronautica et Astronautica Sinica, 2013, 34(9): 2007-2018.(in Chinese) 杨小权, 杨爱明, 孙刚. 一种强耦合Spalart-Allmaras湍流模型的RANS方程的高效数值计算方法[J]. 航空学报, 2013, 34(9): 2007-2018.
[24] Deng X G, Mao M L, Tu G H, et al. Extending weighted compact nonlinear schemes to complex grids with characteristic-based interface conditions[J]. AIAA Journal, 2010, 48(12): 2840-2851.
[25] Jiang Y, Deng X G, Mao M L, et al. Extending seventh-order hybrid cell-edge and cell-node dissipative compact scheme to complex grids//The 4th Asian Symposium on Computational Heat Transfer and Fluid Flow, 2013: 1-10.
[26] Leveque R J. Finite difference methods for ordinary and partial differential equations[M]. Seattle, Washington: SIAM, 2007: 69-110.
[27] Liu X. Research on high-order accurate weighted compact nonlinear scheme and their applications to complicated flows. Mianyang: China Aerodynamics Research and Development Center, 2004. (in Chinese) 刘昕. 高阶精度加权紧致非线性格式研究与其在复杂流动中的应用. 绵阳: 中国空气动力研究与发展中心, 2004.
[28] Berg B V D. Boundary Layer measurements on a two-dimensional wing with flap, NLR-TR-1979-009. Amsterdam, The Nethelands: National Aerospace Laboratory, 1979.
[29] Xu J, Liu Q H, Cai J S, et al. Drag prediction based on overset grids with implicit hole cutting technique[J]. Acta Aeronautica et Astronautica Sinica, 2013, 34(2): 208-217.(in Chinese) 徐嘉, 刘秋洪, 蔡晋生, 等. 基于隐式嵌套重叠网格技术的阻力预测[J]. 航空学报, 2013, 34(2): 208-217.
/
| 〈 |
|
〉 |
CJA