Abstract
The primary impediments impeding the implementation of high-order methods in simulating viscous flow over complex configurations are robustness and convergence. These challenges impose significant constraints on computational efficiency, particularly in the domain of engineering applications. To address these concerns, this paper proposes a robust implicit high-order discontinuous Galerkin (DG) method for solving compressible Navier-Stokes (NS) equations on arbitrary grids. The method achieves a favorable equilibrium between computational stability and efficiency. To solve the linear system, an exact Jacobian matrix solving strategy is employed for preconditioning and matrix-vector generation in the generalized minimal residual (GMRES) method. This approach mitigates numerical errors in Jacobian solution during implicit calculations and facilitates the implementation of an adaptive Courant-Friedrichs-Lewy (CFL) number increasing strategy, with the aim of improving convergence and robustness. To further enhance the applicability of the proposed method for intricate grid distortions, all simulations are performed in the reference domain. This practice significantly improves the reversibility of the mass matrix in implicit calculations. A comprehensive analysis of various parameters influencing computational stability and efficiency is conducted, including CFL number, Krylov subspace size, and GMRES convergence criteria. The computed results from a series of numerical test cases demonstrate the promising results achieved by combining the DG method, GMRES solver, exact Jacobian matrix, adaptive CFL number, and reference domain calculations in terms of robustness, convergence, and accuracy. These analysis results can serve as a reference for implicit computation in high-order calculations.
摘要
为了提高高阶方法在模拟复杂结构粘性流动时的鲁棒性和收敛性, 本文提出了一种隐式高阶间断伽辽金(DG)方法. 该种方法 在计算稳定性和效率之间实现了良好的平衡, 能够有效地处理复杂流动问题. 具体地, 为了求解线性系统, 发展了精确雅可比矩阵求解 方法, 并应用于广义最小残差(GMRES)方法进行预处理和矩阵向量生成. 该方法显著减少了隐式计算中雅可比矩阵的数值误差, 提高 了计算的准确性和稳定性. 同时, 通过自适应CFL数增加策略, 进一步提高了隐式方法的计算效率. 此外, 为了提高所提出方法对复杂网 格畸变的适应性, 所有的模拟都在参数域中进行. 这种方法显著提高了隐式计算中质量矩阵的可逆性, 从而提高了计算的稳定性. 本文 还对影响计算稳定性和效率的各种参数进行了全面分析, 包括CFL数、Krylov子空间大小和GMRES收敛标准. 通过一系列测试算例, 证明了将DG方法、GMRES方法、精确雅可比矩阵计算方法、自适应CFL数和参数域相结合, 能够显著提高计算的鲁棒性、收敛性和 计算精度. 这些分析结果为高阶计算中的隐式计算提供了重要的参考价值.
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References
H. Luo, J.D. Baum, and R. Löhner, A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys. 227, 8875 (2008).
Z.H. Jiang, C. Yan, and J. Yu, Implicit high-order discontinuous Galerkin method with HWENO type limiters for steady viscous flow simulations, Acta Mech. Sin. 29, 526 (2013).
Z.H. Jiang, C. Yan, and J. Yu, A simple a posteriori indicator for discontinuous Galerkin method on unstructured grids, Acta Mech. Sin. 39, 322296 (2023).
P. Delorme, P. Mazet, C. Peyret, and Y. Ventribout, Computational aeroacoustics applications based on a discontinuous Galerkin method, Comptes Rendus Mécanique 333, 676 (2005).
J. Zhao, and H. Tang, Runge-Kutta discontinuous Galerkin methods for the special relativistic magnetohydrodynamics, J. Comput. Phys. 343, 33 (2017).
S. Hennemann, A.M. Rueda-Ramírez, F.J. Hindenlang, and G.J. Gassner, A provably entropy stable subcell shock capturing approach for high order split form DG for the compressible Euler equations, J. Comput. Phys. 426, 109935 (2021).
X. He, K. Wang, T. Liu, Y. Feng, B. Zhang, W. Yuan, and X. Wang, HODG: High-order discontinuous Galerkin methods for solving compressible Euler and Navier-Stokes equations—An open-source component-based development framework, Comput. Phys. Commun. 286, 108660 (2023).
Y. Jiang, and H. Liu, Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations, J. Comput. Phys. 373, 385 (2018).
D.N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal. 19, 742 (1982).
B. Cockburn, and C.W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35, 2440 (1998).
J. Peraire, and P.O. Persson, The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput. 30, 1806 (2008).
B. V. Leer, M. Lo, and M. V. Raalte, in A discontinuous Galerkin method for diffusion based on recovery: Proceedings of the 18th AIAA Computational Fluid Dynamics Conference, Miami, 2007.
H. Luo, L. Luo, R. Nourgaliev, V.A. Mousseau, and N. Dinh, A reconstructed discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, J. Comput. Phys. 229, 6961 (2010).
F. Bassi, A. Crivellini, S. Rebay, and M. Savini, Discontinuous Galerkin solution of the Reynolds-averaged Navier-Stokes and k-ω turbulence model equations, Comput. Fluids 34, 507 (2005).
H. Liu, and J. Yan, The direct discontinuous Galerkin (DDG) methods for diffusion problems, SIAM J. Numer. Anal. 47, 675 (2009).
H. Liu, and J. Yan, The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections, Commun. Comput. Phys. 8, 541 (2010).
J. Cheng, X. Yang, X. Liu, T. Liu, and H. Luo, A direct discontinuous Galerkin method for the compressible Navier-Stokes equations on arbitrary grids, J. Comput. Phys. 327, 484 (2016).
J. Cheng, X. Liu, X. Yang, T. Liu, and H. Luo, in A direct discontinuous Galerkin method for computation of turbulent flows on hybrid grids: Proceedings of the 46th AIAA Fluid Dynamics Conference, Washington, 2016.
J. Jaśkowiec, Discontinuous Galerkin method on reference domain, Comput. Assist. Methods Eng. Sci. 22, 177 (2017).
J. Jaśkowiec, Very high order discontinuous Galerkin method in elliptic problems, Comput. Mech. 62, 1 (2018).
H. Luo, J.D. Baum, and R. Löhner, A fast, matrix-free implicit method for compressible flows on unstructured grids, J. Comput. Phys. 146, 664 (1998).
H. Luo, H. Segawa, and M.R. Visbal, An implicit discontinuous Galerkin method for the unsteady compressible Navier-Stokes equations, Comput. Fluids 53, 133 (2012).
H. Ying, and L. Hao, Preconditioned GMRES method for a class of Toeplitz linear systems, Math. Numer. Sin. 43, 177 (2021).
S. Correnty, E. Jarlebring, and K. M. Soodhalter, Preconditioned infinite GMRES for parameterized linear systems, SIAM J. Sci. Comput. S120 (2023).
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed. (Soc. for Industrial and Applied Mathematics, Philadelphia, 2003).
F. Bassi, and S. Rebay, GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations, in: Discontinuous Galerkin Methods (Springer, Berlin, Heidelberg, 2000), pp. 197–208.
P.O. Persson, and J. Peraire, Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier-Stokes equations, SIAM J. Sci. Comput. 30, 2709 (2008).
M. J. Zahr, and P. O. Persson, in Performance tuning of Newton-GMRES methods for discontinuous Galerkin discretizations of the Navier-Stokes equations: Proceedings of the 21st AIAA Computational Fluid Dynamics Conference, San Diego, 2013.
T.L. Tysinger, and D.A. Caughey, Alternating direction implicit methods for the Navier-Stokes equations, AIAA J. 30, 2158 (1992).
J. Liu, J. Chen, Z. Zhang, Y. Yang, and Z. Xiao, Assessment ofa new hybrid-SSOR implicit temporal scheme for turbulent flows across a wide range of Mach numbers, Acta Mech. Sin. 39, 322398 (2023).
N. Nigro, M. Storti, S. Idelsohn, and T. Tezduyar, Physics based GMRES preconditioner for compressible and incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng. 154, 203 (1998).
X. Yang, C. Jian, C. Wang, and H. Luo, in A fast, implicit discontinuous Galerkin method based on analytical Jacobians for the compressible Navier-Stokes equations: Proceedings of the 54th AIAA Aerospace Sciences Meeting, San Diego, 2016.
X. Yang, J. Cheng, H. Luo, and Q. Zhao, Robust implicit direct discontinuous Galerkin method for simulating the compressible turbulent flows, AIAA J. 57, 1113 (2019).
S.C. Eisenstat, and H.F. Walker, Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput. 17, 16 (1996).
H.B. An, Z.Y. Mo, and X.P. Liu, A choice of forcing terms in inexact Newton method, J. Comput. Appl. Math. 200, 47 (2007).
K. Lund, Adaptively restarted block Krylov subspace methods with low-synchronization skeletons, Numer. Algor. 93, 731 (2023).
S. R. Allmaras, F. T. Johnson, and P. R. Spalart, in Modifications and clarifications for the implementation of the Spalart-Allmaras turbulence model: Proceedings of the 7th International Conference on Computational Fluid Dynamics, Big Island, Hawaii, 2012.
P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43, 357 (1981).
H. Liu, Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations, Math. Comput. 84, 2263 (2015).
Q. Zou, GMRES algorithms over 35 years, Appl. Math. Comput. 445, 127869 (2023).
W. Cao, H. Liu, and Z. Zhang, Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations, Numer. Meth. Part. D. E. 33, 290 (2017).
T. Poinsot, and S.M. Candel, The influence of differencing and CFL number on implicit time-dependent non-linear calculations, J. Comput. Phys. 62, 282 (1986).
T. Warburton, and T. Hagstrom, Taming the CFL number for discontinuous Galerkin methods on structured meshes, SIAM J. Numer. Anal. 46, 3151 (2008).
S. Joshi, J. Kou, A. Hurtado de Mendoza, K. Puri, C. Hirsch, G. Rubio, and E. Ferrer, Length-scales for efficient CFL conditions in high-order methods with distorted meshes: Application to local-timestepping for p-multigrid, Comput. Fluids 265, 106011 (2023).
Y.H. Tseng, and J.H. Ferziger, A ghost-cell immersed boundary method for flow in complex geometry, J. Comput. Phys. 192, 593 (2003).
NASA, Turbulence Modeling Resource, https://turbmodels.larc.nasa.gov/.
M. Murayama, Y. Yokokawa, H. Ura, K. Nakakita, K. Yamamoto, Y. Ito, T. Takaishi, R. Sakai, K. Shimoda, T. Kato, and T. Homma, in Experimental study of slat noise from 30P30N three-element high-lift airfoil in JAXA Kevlar-wall low-speed wind tunnel: Proceedings of the AIAA/CEAS Aeroacoustics Conference, Atlanta, 2018.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 12102247) and the Technology Development Program (Grant No. JCKY2022110C119).
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Author contributions Jia Yan carried out the work and wrote the first draft of the manuscript. Xiaoquan Yang helped organize the manuscript. Peifen Weng revised and edited the final version.
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Yan, J., Yang, X. & Weng, P. A robust implicit high-order discontinuous Galerkin method for solving compressible Navier-Stokes equations on arbitrary grids. Acta Mech. Sin. 40, 323429 (2024). https://doi.org/10.1007/s10409-024-23429-x
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DOI: https://doi.org/10.1007/s10409-024-23429-x