Multigrid algorithms for high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations.(English)Zbl 1391.65181

Summary: Multigrid algorithms are developed for systems arising from high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations on unstructured meshes. The algorithms are based on coupling both $$p$$- and $$h$$-multigrid ($$ph$$-multigrid) methods which are used in nonlinear or linear forms, and either directly as solvers or as preconditioners to a Newton-Krylov method.
The performance of the algorithms are examined in solving the laminar flow over an airfoil configuration. It is shown that the choice of the cycling strategy is crucial in achieving efficient and scalable solvers. For the multigrid solvers, while the order-independent convergence rate is obtained with a proper cycle type, the mesh-independent performance is achieved only if the coarsest problem is solved to a sufficient accuracy. On the other hand, the multigrid preconditioned Newton-GMRES solver appears to be insensitive to this condition and mesh-independent convergence is achieved under the desirable condition that the coarsest problem is solved using a fixed number of multigrid cycles regardless of the size of the problem.
It is concluded that the Newton-GMRES solver with the multigrid preconditioning yields the most efficient and robust algorithm among those studied.

MSC:

 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65F08 Preconditioners for iterative methods 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 76D05 Navier-Stokes equations for incompressible viscous fluids

HLLE; Wesseling
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 [1] Hartmann, R.; Houston, P., Symmetric interior penalty DG methods for the compressible navier – stokes equations I: method formulations, Int. J. numer. anal. model., 3, 1-20, (2006) · Zbl 1129.76030 [2] Klaij, C.M.; van der Vegt, J.J.W.; van der Ven, H., Space-time discontinuous Galerkin method for the compressible navier – stokes equations, J. comput. phys., 217, 589-611, (2006) · Zbl 1099.76035 [3] Davis, S.F., Simplified second-order Godunov-type methods, SIAM J. sci. stat. comput., 9, 445-473, (1988) · Zbl 0645.65050 [4] Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes, J. comput. phys., 43, 357-372, (1981) · Zbl 0474.65066 [5] Harten, A.; Lax, P.D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35-61, (1983) · Zbl 0565.65051 [6] Toro, F.E., Riemann solvers and numerical methods for fluid dynamics, Appl. mech., (1999), Springer New York, NY · Zbl 0923.76004 [7] Batten, P.; Clarke, N.; Lambert, C.; Causon, D.M., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. sci. comput., 18, 1553-1570, (1997) · Zbl 0992.65088 [8] D.J. Mavriplis, Unstructured mesh generation and adaptivity, in: VKI Lecture Series VKI-LS 1995-02, 1995. [9] Mavriplis, D.J.; Venkatakrishnan, V., Agglomeration multigrid for two-dimensional viscous flows, Comput. fluids, 24, 5, 553-570, (1995) · Zbl 0846.76047 [10] Mavriplis, D.J., An assessment of linear vs. non-linear multigrid methods for unstructured mesh solvers, J. comput. phys., 175, 302-325, (2002) · Zbl 0995.65099 [11] Wesselling, P., An introduction to multigrid methods, (1992), Willy New York, p. 284 [12] Lötstedt, P., Grid independent convergence of the multigrid method for first-order equations, J. SIAM numer. anal., 29, 1370-1394, (1992) · Zbl 0759.65083 [13] Fidkowski, K.J.; Oliver, T.A.; Lu, J.; Darmofal, D.L., p-multigrid solution of high-order discontinuous Galerkin discretizations of the compressible navier – stokes equations, J. comput. phys., 207, 92-113, (2005) · Zbl 1177.76194 [14] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 856-869, (1986) · Zbl 0599.65018 [15] Saad, Y., Iterative methods for sparse linear system, (1996), PWS Press New York [16] B.T. Helenbrook, D.J. Mavriplis, H.A. Atkins, Analysis of p-multigrid for continuous and discontinuous finite element discretizations, AIAA Paper 2003-3989, 2003. [17] Jameson, A., Solution of the Euler equations for two-dimensional transonic flow by a multigrid method, Appl. math. comput., 13, 327-356, (1983) · Zbl 0545.76065 [18] Mavriplis, D.J., Multigrid solution of the 2-D Euler equations on unstructured triangular meshes, Aiaa j., 26, 824-831, (1988) · Zbl 0667.76088 [19] Mavriplis, D.J., Multigrid strategies for viscous flow solvers on anisotropic unstructured meshes, J. comput. phys., 145, 141-165, (1998) · Zbl 0926.76066 [20] Nastase, C.; Mavriplis, D.J., High-order discontinuous Galerkin methods using an hp-multigrid approach, J. comput. phys., 213, 330-357, (2006) · Zbl 1089.65100 [21] C. Nastase, D.J. Mavriplis, A parallel hp-Multigrid solver for three-dimensional discontinuous Galerkin discretizations of the Euler equations, AIAA Paper 2007-512, 2007. [22] Lottes, J.W.; Fischer, P.F., Hybrid multigrid/Schwarz algorithms for the spectral element method, J. sci. comput., 24, 45-78, (2005) · Zbl 1078.65570 [23] Rönquist, E.M.; Patera, A.T., Spectral element multigrid. I. formulation and numerical results, J. sci. comput., 2, 389-406, (1987) · Zbl 0666.65055 [24] Hackbusch, W., Multigrid methods and applications, (1995), Springer Berlin [25] Shahbazi, K., An explicit expression for the penalty parameter of the interior penalty method, J. comput. phys., 205, 401-407, (2005) · Zbl 1072.65149 [26] Solin, P.; Segeth, P.; Zel, I., High-order finite element methods, Studies in advanced mathematics, (2003), Chapman and Hall London [27] Arnold, D., An interior penalty finite element method with discontinuous elements, SIAM J. numer. anal., 19, 742-760, (1982) · Zbl 0482.65060 [28] Kelly, C.T.; Keyes, D.E., Convergence analysis of pseudo-transient continuation, SIAM J. numer. anal., 35, 508-523, (1998) · Zbl 0911.65080 [29] Persson, P.O.; Peraire, J., Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the navier – stokes equations, SIAM J. sci. comput., 30, 2709-2733, (2008) · Zbl 1362.76052 [30] Thomas, L.W., Numerical partial differential equations, (1995), Springer [31] Valarezo, W.O.; Mavriplis, D.J., Navier – stokes applications to high-lift airfoil analysis, AIAA J. aircraft, 23, 457-688, (1995)
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