基于雅可比矩阵精确计算的GMRES隐式方法在间断Galerkin有限元中的应用
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摘要
为改善高阶间断Galerkin有限元方法(DG)时间推进效率,在三维非结构网格下针对该方法建立了并行广义最小残差(Generalized Minimal Residual,GMRES)隐式时间迭代方法,GMRES方法基于科学计算工具包PETSc中的Krylov子空间求解器实现。为进一步提高GMRES的计算效率,发展了方程组右端项残值雅可比精确计算方法,针对无黏通量Roe格式和黏性通量BR2(Bassi Rebay 2)黏性计算方法,分别解析给出其对守恒变量多项式自由度的雅可比矩阵。基于建立的方法首先采用NACA0012翼型研究了GMRES的重启次数及收敛参数对方法收敛性影响,然后采用无黏及黏性算例对比研究了基于雅可比矩阵不同计算方法的GMRES计算效率,同时对比研究了雅可比矩阵完全近似求解下GMRES和LU-SGS(Lower Upper-Symmetric Gauss-Seidel)的计算效率。结果表明,建立的基于右端项残值雅可比矩阵精确求解的GMRES方法能够大幅提高不同精度DG方法的CFL(CourantFriedrichs-Lewy)数,相比前面提到的其它方法具有更高的计算效率,其收敛速度实现量级以上的提高。
To improve iteration efficiency of high-order discontinuous Galerkin finite element(DG)method,an efficient implicit generalized minimal residual(GMRES)method has been applied to DG method based on unstructured grids.The method is implemented by the ksp solver of portable,extensible toolkit for scientific computation(PETSc)library.In order to further improve computational efficiency of GMRES,the exact calculation method of Jacobian matrix is developed.The Jacobian matrix of Roe scheme and bassi rebay 2(BR2)scheme are exactly obtained for the freedom of conservation variables.The developed method is applied to compute a variety of steady inviscid and viscous flow problems.First,a NACA0012 airfoil is used to study the effect of restarted iteration and convergence parameters on the GMRES convergence.The computational efficiencies of the GMRES based on different calculation methods of Jacobian matrix are compared by the inviscid and viscous examples.The computational efficiencies are also compared for the GMRES and LU-SGS iterative method.The results show that the exact calculation method of Jacobian matrix can greatly increase the CFL number of the GMRES for different DG methods.Compared with other methods,the GMRES based on the exact calculation method of Jacobian matrix has higher computational efficiency,and its convergence speed is improved by more than one order of magnitude.
To improve iteration efficiency of high-order discontinuous Galerkin finite element(DG)method,an efficient implicit generalized minimal residual(GMRES)method has been applied to DG method based on unstructured grids.The method is implemented by the ksp solver of portable,extensible toolkit for scientific computation(PETSc)library.In order to further improve computational efficiency of GMRES,the exact calculation method of Jacobian matrix is developed.The Jacobian matrix of Roe scheme and bassi rebay 2(BR2)scheme are exactly obtained for the freedom of conservation variables.The developed method is applied to compute a variety of steady inviscid and viscous flow problems.First,a NACA0012 airfoil is used to study the effect of restarted iteration and convergence parameters on the GMRES convergence.The computational efficiencies of the GMRES based on different calculation methods of Jacobian matrix are compared by the inviscid and viscous examples.The computational efficiencies are also compared for the GMRES and LU-SGS iterative method.The results show that the exact calculation method of Jacobian matrix can greatly increase the CFL number of the GMRES for different DG methods.Compared with other methods,the GMRES based on the exact calculation method of Jacobian matrix has higher computational efficiency,and its convergence speed is improved by more than one order of magnitude.
引文
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[5]TAKANORI H,KEISUKE S,WANG Z J.Animplicit LU-SGS scheme for the spectral volume method on unstructured tetrahedral grids[J].Communications in Computational Physics,2009,6(5):978-996.
[6]刘伟,张来平,赫新等.基于Newton/Gauss-Seidel迭代的DGM隐式方法[J].力学学报,2012,44(4):792-796.LIU W,ZHANG L P,HE X,et al.An implicit algorithm for discontinuous Galerkin method based on Newton/Gauss-seidel iterations[J].Chinese Journal of Theoretical and Applied Mechanics,2012,44(4):792-796.(in Chinese)
[7]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.
[8]CRIVELLINI A,BASSI F.An implicit matrix-free discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations[J].Computers&Fluids,2011,50:81-93.
[9]LASLO T D,DAVID L D.Discontinuous Galerkin solutions of the Navier-Stokes equations using linear multigrid preconditioning[R].AIAA 2007-3942.
[10]CHEN S Q,ZHANG T T.Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes:application to discontinuous Galerkin methods[J].Journal of Computational Physics,2011,230:4336-4352.
[11]LANDMANN B.A parallel discontinuous Galerkin code for the Navier-Stokes and Reynolds-averaged Navier-Stokes equations[D].Stuttgart,university of Stuttgart,2008.
[12]党亚斌,刘凯礼,孙一峰,等.用隐式高精度间断伽辽金方法模拟可压层流和湍流[J].空气动力学学报,2018,36(3):535-541.DANG Y B,LIU K L,SUN Y F,et al.Implicit high order discontinuous Galerkin method for compressible laminar and turbulent flow simulation[J].Acta Aerodynamica Sinica,2018,36(3):535-541.(in Chinese)
[13]YANG X Q,CHENG J,WANG C J,et al.A fast,implicit discontinuous Galerkin method based on analytical Jacobians for the compressible Navier-Stokes equations[R].AIAA 2016-1326.
[14]CHENG J,LIU X D,YANG X Q,et al.A direct discontinuous Galerkin method for the compressible NavierStokes equations on arbitrary grids[R].AIAA 2016-3334.
[15]BALAY S,ABHYANKAR S,ADAMS M,et al.PETSc users manual[OL].http://www.mcs.anl.gov/petsc/documentation/index.html.
[16]BLAZEK J.Computational fluid dynamics:principles and applications[M].London Elsevier Science,2001.
[17]ROE P L.Approximate riemann solvers,parameter vectors,and difference schemes[J].Journal of Computational Physics,1981,43:357-372.
[18]BASSI F,REBAY S,MARIOTTI G,et al.A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows[C]//In Proceedings of the 2nd European Conference on Turbomachinery Fluid Dynamics and Thermodynamics,March 5-7 1997,99-108.
[19]KRIST S L,BIEDRON R T,RUMSEY C L.CFL3D User’s Manual(Version 5.0)[EB].NASA/TM-1998-208444.
[20]SCHMITT V,CHARPIN F.Pressure distributions on the ONERA-M6 wing at transonic Mach numbers[R].Experimental Data Base for Computer Program Assessment,AGARD AR138,1979.