×

Second derivative time integration methods for discontinuous Galerkin solutions of unsteady compressible flows. (English) Zbl 1380.76131

Summary: In this paper, we investigate the possibility of using the high-order accurate \(A(\alpha)\)-stable second derivative (SD) schemes proposed by Enright for the implicit time integration of the discontinuous Galerkin (DG) space-discretized Navier-Stokes equations. These multistep schemes are A-stable up to fourth-order, but their use results in a system matrix difficult to compute. Furthermore, the evaluation of the nonlinear function is computationally very demanding. Here, we propose a matrix-free (MF) implementation of Enright schemes that allows to obtain a method without the costs of forming, storing and factorizing the system matrix, which is much less computationally expensive than its matrix-explicit counterpart, and which performs competitively with other implicit schemes, such as the modified extended backward differentiation formulae (MEBDF). The algorithm makes use of the preconditioned GMRES algorithm for solving the linear system of equations. The preconditioner is based on the ILU(0) factorization of an approximated but computationally cheaper form of the system matrix, and it has been reused for several time steps to improve the efficiency of the MF Newton-Krylov solver. We additionally employ a polynomial extrapolation technique to compute an accurate initial guess to the implicit nonlinear system. The stability properties of SD schemes have been analyzed by solving a linear model problem. For the analysis on the Navier-Stokes equations, two-dimensional inviscid and viscous test cases, both with a known analytical solution, are solved to assess the accuracy properties of the proposed time integration method for nonlinear autonomous and non-autonomous systems, respectively. The performance of the SD algorithm is compared with the ones obtained by using an MF-MEBDF solver, in order to evaluate its effectiveness, identifying its limitations and suggesting possible further improvements.

MSC:

76N15 Gas dynamics (general theory)
76M10 Finite element methods applied to problems in fluid mechanics
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] (Cockburn, B.; Karniadakis, G. E.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation, and Applications. Discontinuous Galerkin Methods: Theory, Computation, and Applications, Lect. Notes Comput. Sci. Eng., vol. 11 (2000), Springer) · Zbl 0935.00043
[2] Bassi, F.; Rebay, S., A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations, J. Comput. Phys., 131, 267-279 (1997) · Zbl 0871.76040
[3] Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2002) · Zbl 1008.65080
[4] Persson, P.-O.; Peraire, J., An Efficient Low Memory Implicit DG Algorithm for Time Dependent Problems (2006), AIAA Paper 113:2006
[5] Dolejsi, V.; Kus, P., Adaptive backward difference formula - discontinuous Galerkin finite element method for the solution of conservation laws, Int. J. Numer. Methods Eng., 73, 12, 1739-1766 (2008) · Zbl 1159.76349
[6] Wang, L.; Mavriplis, D. J., Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations, J. Comput. Phys., 225, 1994-2015 (2007) · Zbl 1343.76022
[7] Persson, P.-O., High-order LES simulations using implicit-explicit Runge-Kutta schemes, (Proceedings of the 49th AIAA Aerospace Sciences Meeting and Exhibit. Proceedings of the 49th AIAA Aerospace Sciences Meeting and Exhibit, AIAA, vol. 684 (2011))
[8] Birken, P.; Gassner, G.; Haas, M.; Munz, C.-D., Preconditioning for modal discontinuous Galerkin methods for unsteady 3D Navier-Stokes equations, J. Comput. Phys., 240, 20-35 (2013) · Zbl 1426.76520
[9] Birken, P.; Gassner, G.; Haas, M.; Munz, C.-D., Efficient time integration for discontinuous Galerkin method for unsteady 3D Navier-Stokes equations, (ECCOMAS 2012. ECCOMAS 2012, Vienna, Austria, September 10-14 (2012)), 1-20
[10] Nigro, A.; Ghidoni, A.; Rebay, S.; Bassi, F., Modified extended BDF scheme for the discontinuous Galerkin solution of unsteady compressible flows, Int. J. Numer. Methods Fluids, 76, 9, 549-574 (2014)
[11] Nigro, A.; De Bartolo, C.; Bassi, F.; Ghidoni, A., Up to sixth-order accurate A-stable implicit schemes applied to the discontinuous Galerkin discretized Navier-Stokes equations, J. Comput. Phys., 276, 136-162 (2014) · Zbl 1349.76247
[12] Bassi, F.; Crivellini, A.; Di Pietro, D. A.; Rebay, S., An implicit high-order discontinuous Galerkin method for steady and unsteady incompressible flows, Comput. Fluids, 36, 10, 1529-1546 (2007) · Zbl 1194.76102
[13] Nigro, A.; Renda, S.; De Bartolo, C.; Hartmann, R.; Bassi, F., A high-order accurate discontinuous Galerkin finite element method for laminar low Mach number flows, Int. J. Numer. Methods Fluids, 72, 1, 43-68 (2013) · Zbl 1455.76163
[14] Bassi, F.; Botti, L.; Colombo, A.; Ghidoni, A.; Massa, F., Linearly implicit Rosenbrock-type Runge-Kutta schemes applied to the discontinuous Galerkin solution of compressible and incompressible unsteady flows, Comput. Fluids, 118, 305-320 (2015) · Zbl 1390.76833
[15] Noventa, G.; Massa, F.; Bassi, F.; Colombo, A.; Franchina, N.; Ghidoni, A., A high-order discontinuous Galerkin solver for unsteady incompressible turbulent flows, Comput. Fluids, 139, 248-260 (2016) · Zbl 1390.76344
[16] Dahlquist, G., A special stability problem for linear multistep methods, BIT Numer. Math., 3, 1, 27-43 (1963) · Zbl 0123.11703
[17] Enright, W. H., Second derivative multistep methods for Stiff ordinary differential equations, SIAM J. Numer. Anal., 11, 2, 321-331 (1974) · Zbl 0249.65055
[18] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 2, 357-397 (2004) · Zbl 1036.65045
[19] Crivellini, A.; Bassi, F., An implicit matrix-free discontinuous Galerkin solver for viscous and turbulent aerodynamic simulations, Comput. Fluids, 50, 1, 81-93 (2011) · Zbl 1271.76164
[20] Tesini, P., An h-Multigrid Approach for High-Order Discontinuous Galerkin Methods (January 2008), University of Bergamo: University of Bergamo Bergamo, Italy, Ph.D. thesis
[21] Bassi, F.; Botti, L.; Colombo, A.; Di Pietro, D. A.; Tesini, P., On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations, J. Comput. Phys., 231, 1, 45-65 (2012) · Zbl 1457.65178
[22] Di Pietro, D. A.; Ern, A., Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques et Applications, vol. 69 (2012), Springer · Zbl 1231.65209
[23] Bassi, F.; Rebay, S.; Mariotti, G.; Pedinotti, S.; Savini, M., A high-order accurate discontinuous finite element method for inviscid and viscous turbomachinery flows, (Decuypere, R.; Dibelius, G., Proceedings of the 2nd European Conference on turbomachinery Fluid Dynamics and Thermodynamics. Proceedings of the 2nd European Conference on turbomachinery Fluid Dynamics and Thermodynamics, Antwerpen, Belgium, March 5-7 (1997)), 99-108
[24] Bassi, F.; Rebay, S., A high order discontinuous Galerkin method for compressible turbulent flows, (Cockburn, B.; Karniadakis, G.; Shu, C.-W., Discontinuous Galerkin Methods: Theory, Computation and Applications. Discontinuous Galerkin Methods: Theory, Computation and Applications, Lect. Notes Comput. Sci. Eng., vol. 11 (2000), Springer), 77-88 · Zbl 0991.76039
[25] Brezzi, F.; Manzini, M.; Marini, D.; Pietra, P.; Russo, A., Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differ. Equ., 16, 4, 365-378 (2000) · Zbl 0957.65099
[26] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (1996), Springer-Verlag · Zbl 0859.65067
[27] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K.; Dalcin, L.; Eijkhout, V.; Gropp, W. D.; Kaushik, D.; Knepley, M. G.; McInnes, L. C.; Rupp, K.; Smith, B. F.; Zampini, S.; Zhang, H. (2015), Petsc web-page
[28] Pernice, M.; Walker, H. F., NITSOL: a Newton iterative solver for nonlinear systems, SIAM J. Sci. Comput., 19, 1, 302-318 (1998) · Zbl 0916.65049
[29] Nigro, A.; De Bartolo, C.; Crivellini, A.; Bassi, F., Matrix-free modified extended backward differentiation formulae applied to the discontinuous Galerkin solution of compressible unsteady viscous flows, (Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering. Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2016, Crete Island, Greece, June 5-10 (2016)), 1-18
[30] Cash, J. R., The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae, Comput. Math. Appl., 5, 9, 645-657 (1983) · Zbl 0526.65052
[31] Bassi, F.; Botti, L.; Colombo, A.; Crivellini, A.; Ghidoni, A.; Nigro, A.; Rebay, S., Time integration in the discontinuous Galerkin code MIGALE - unsteady problems, (Kroll, N.; Hirsch, C.; Bassi, F.; Johnston, C.; Hillewaert, K., IDIHOM: Industrialization of High-Order Methods - A Top-Down Approach. IDIHOM: Industrialization of High-Order Methods - A Top-Down Approach, Notes Numer. Fluid Mech. Multidiscipl. Des., vol. 128 (2015), Springer), 205-230
[32] Nigro, A.; De Bartolo, C.; Bassi, F.; Ghidoni, A., High-order discontinuous Galerkin solution of unsteady flows by using an advanced implicit method, (High Order Nonlinear Numerical Schemes for Evolutionary PDEs. High Order Nonlinear Numerical Schemes for Evolutionary PDEs, Lect. Notes Comput. Sci. Eng., vol. 99 (2014)), 135-149 · Zbl 1371.76099
[33] Hu, C.; Shu, C. W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 1, 97-127 (1999) · Zbl 0926.65090
[34] Gottlieb, J. J.; Groth, C. P.T., Assessment of Riemann solvers for unsteady one-dimensional inviscid flows of perfect gases, J. Comput. Phys., 78, 2, 437-458 (1988) · Zbl 0657.76064
[35] Salari, K.; Knupp, P., Code Verification by the Method of Manufactured Solutions (2000), Sandia National Laboratories: Sandia National Laboratories Albuquerque, NM, SAND 2000-1444
[36] Mathematica, Version 7.0 (2008), Wolfram Research, Inc.: Wolfram Research, Inc. Champaign, IL
[37] Hanel, D.; Schwane, R.; Seider, G., On the Accuracy of Upwind Schemes for the Solution of the Navier-Stokes Equations (1987), AIAA Paper 87-1105
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.