×

Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier-Stokes equations. (English) Zbl 1426.76520

Summary: We compare different block preconditioners in the context of parallel time adaptive higher order implicit time integration using Jacobian-free Newton-Krylov (JFNK) solvers for discontinuous Galerkin (DG) discretizations of the three-dimensional time-dependent Navier-Stokes equations. A special emphasis of this work is the performance for a relative high number of processors, i.e. with a low number of elements on the processor. For high order DG discretizations, a particular problem that needs to be addressed is the size of the blocks in the Jacobian. Thus, we propose a new class of preconditioners that exploits the hierarchy of modal basis functions and introduces a flexible order of the off-diagonal Jacobian blocks. While the standard preconditioners ‘block Jacobi’ (no off-blocks) and full symmetric Gauss-Seidel (full off-blocks) are included as special cases, the reduction of the off-block order results in the new scheme ROBO-SGS. This allows us to investigate the impact of the preconditioner’s sparsity pattern with respect to the computational performance. Since the number of iterations is not well suited to judge the efficiency of a preconditioner, we additionally consider CPU time for the comparisons. We found that both block Jacobi and ROBO-SGS have good overall performance and good strong parallel scaling behavior.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76D05 Navier-Stokes equations for incompressible viscous fluids
PDFBibTeX XMLCite
Full Text: DOI

References:

[2] Bassi, F.; Ghidoni, A.; Rebay, S., Optimal Runge-Kutta smoothers for the \(p\)-multigrid discontinuous Galerkin solution of the 1D Euler equations, J. Comput. Phys., 11, 4153-4175 (2011) · Zbl 1220.65130
[3] Bassi, F.; Ghidoni, A.; Rebay, S.; Tesini, P., High-order accurate \(p\)-multigrid discontinuous Galerkin solution of the Euler equations, Int. J. Numer. Methods Fluids, 60, 847-865 (2009) · Zbl 1165.76022
[4] Bassi, F.; Rebay, S., A high-order discontinuous Galerkin finite element method solution of the 2D Euler equations, J. Comput. Phys., 138, 251-285 (1997) · Zbl 0902.76056
[6] Birken, P., Solving nonlinear systems inside implicit time integration schemes for unsteady viscous flows, (Ansorge, R.; Bijl, H.; Meister, A.; Sonar, T., Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws (2013), Springer), 57-71 · Zbl 1381.76253
[7] Cockburn, B.; Shu, C.-W., The local discontinuous Galerkin method for time-dependent convection diffusion systems, SIAM J. Numer. Anal., 35, 2440-2463 (1998) · Zbl 0927.65118
[8] Dembo, R.; Eisenstat, R.; Steihaug, T., Inexact Newton methods, SIAM J. Numer. Anal., 19, 400-408 (1982) · Zbl 0478.65030
[9] Diosady, L. T.; Darmofal, D. L., Preconditioning methods for discontinuous Galerkin solutions of the Navier-Stokes equations, J. Comput. Phys., 228, 3917-3935 (2009) · Zbl 1185.76812
[10] Dolejší, V.; Feistauer, M., A semi-implicit discontinuous Galerkin finite element method for the numerical solution of inviscid compressible flow, J. Comput. Phys., 198, 727-746 (2004) · Zbl 1116.76386
[11] Eisenstat, S. C.; Walker, H. F., Choosing the forcing terms in an inexact Newton method, SIAM J. Sci. Comput., 17, 16-32 (1996) · Zbl 0845.65021
[12] 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
[13] Gassner, G.; Lörcher, F.; Munz, C.-D., A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes, J. Comput. Phys., 224, 1049-1063 (2007) · Zbl 1123.76040
[14] Gassner, G.; Lörcher, F.; Munz, C.-D., A discontinuous Galerkin scheme based on a space-time expansion II. Viscous flow equations in multi dimensions, J. Sci. Comput., 34, 260-286 (2008) · Zbl 1218.76027
[15] Gassner, G. J.; Lörcher, F.; Munz, C.-D.; Hesthaven, J. S., Polymorphic nodal elements and their application in discontinuous Galerkin methods, J. Comput. Phys., 228, 5, 1573-1590 (2009) · Zbl 1267.76062
[16] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II. Solving Ordinary Differential Equations II, Series in Computational Mathematics, vol. 14 (2004), Springer: Springer Berlin
[17] Hesthaven, J. S.; Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (2008), Springer · Zbl 1134.65068
[18] Hindenlang, Florian; Gassner, Gregor J.; Altmann, Christoph; Beck, Andrea; Staudenmaier, Marc; Munz, Claus-Dieter, Explicit discontinuous Galerkin methods for unsteady problems, Comput. Fluids, 61, 0, 86-93 (2012) · Zbl 1365.76117
[19] Jeong, J.; Hussain, F., On the identification of a vortex, J. Fluid Mech., 285, 69-94 (1995) · Zbl 0847.76007
[20] Jothiprasad, G.; Mavriplis, D. J.; Caughey, D. A., Higher-order time integration schemes for the unsteady Navier-Stokes equations on unstructured meshes, J. Comput. Phys., 191, 542-566 (2003) · Zbl 1134.76428
[21] Kanevsky, A.; Carpenter, M. H.; Gottlieb, D.; Hesthaven, J. S., Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes, J. Comput. Phys., 225, 1753-1781 (2007) · Zbl 1123.65097
[22] Kennedy, C. A.; Carpenter, M. H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 139-181 (2003) · Zbl 1013.65103
[23] Klaij, C. M.; van Raalte, M. H.; van der Vegt, J. J.W.; van der Ven, H., \(h\)-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier-Stokes equations, J. Comput. Phys., 227, 1024-1045 (2007) · Zbl 1126.76031
[24] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-397 (2004) · Zbl 1036.65045
[25] Kopriva, D. A.; Woodruff, S. L.; Hussaini, M. Y., Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method, Int. J. Numer. Methods Eng., 53, 105-122 (2002) · Zbl 0994.78020
[27] Lörcher, F.; Gassner, G.; Munz, C.-D., An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations, J. Comput. Phys., 227, 11, 5649-5670 (2008) · Zbl 1147.65077
[28] May, G.; Iacono, F.; Jameson, A., A hybrid multilevel method for high-order discretization of the Euler equations on unstructured meshes, J. Comput. Phys., 229, 3938-3956 (2010) · Zbl 1425.65103
[29] McHugh, P. R.; Knoll, D. A., Comparison of standard and matrix-free implementations of several Newton-Krylov solvers, AIAA J., 32, 12, 2394-2400 (1994) · Zbl 0832.76071
[30] Meister, A.; Vömel, C., Efficient preconditioning of linear systems arising from the discretization of hyperbolic conservation laws, Adv. Comput. Math., 14, 1, 49-73 (2001) · Zbl 0977.65034
[31] Nastase, C. R.; Mavriplis, D. J., High-order discontinuous Galerkin methods using an hp-multigrid approach, J. Comput. Phys., 213, 330-357 (2006) · Zbl 1089.65100
[32] Peraire, J.; Persson, P.-O., The compact discontinuous Galerkin (CDG) method for elliptic problems, SIAM J. Sci. Comput., 30, 4, 1806-1824 (2008) · Zbl 1167.65436
[33] 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
[35] Qin, N.; Ludlow, D. K.; Shaw, S. T., A matrix-free preconditioned Newton/GMRES method for unsteady Navier-Stokes solutions, Int. J. Numer. Methods Fluids, 33, 223-248 (2000) · Zbl 0976.76049
[36] Rasetarinera, P.; Hussaini, M. Y., An efficient implicit discontinuous spectral Galerkin method, J. Comput. Phys., 172, 718-738 (2001) · Zbl 0986.65093
[37] Renac, F.; Marmignon, C.; Coquel, F., Time implicit high-order discontinuous Galerkin method with reduced evaluation cost, SIAM J. Sci. Comput., 34, 1, A370-A394 (2012) · Zbl 1241.65085
[38] 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
[39] Söderlind, G.; Wang, L., Evaluating numerical ODE/DAE methods, algorithms and software, J. Comput. Appl. Math., 185, 244-260 (2006) · Zbl 1081.65533
[40] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1999), Springer · Zbl 0923.76004
[41] Venkatakrishnan, V.; Mavriplis, D. J., Implicit solvers for unstructured meshes, J. Comput. Phys., 105, 1, 83-91 (1993) · Zbl 0783.76065
[42] Vincent, P. E.; Jameson, A., Facilitating the adoption of unstructured high-order methods amongst a wider community of fluid dynamicists, Math. Model. Nat. Phenom., 6, 3, 97-140 (2011) · Zbl 1387.76002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.