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A coupled implicit-explicit time integration method for compressible unsteady flows. (English) Zbl 1453.76104

Summary: This paper addresses how two time integration schemes, the Heun’s scheme for explicit time integration and the second-order Crank-Nicolson scheme for implicit time integration, can be coupled spatially. This coupling is the prerequisite to perform a coupled Large Eddy Simulation/Reynolds Averaged Navier-Stokes computation in an industrial context, using the implicit time procedure for the boundary layer (RANS) and the explicit time integration procedure in the LES region. The coupling procedure is designed in order to switch from explicit to implicit time integrations as fast as possible, while maintaining stability. After introducing the different schemes, the paper presents the initial coupling procedure adapted from a published reference and shows that it can amplify some numerical waves. An alternative procedure, studied in a coupled time/space framework, is shown to be stable and with spectral properties in agreement with the requirements of industrial applications. The coupling technique is validated with standard test cases, ranging from one-dimensional to three-dimensional flows.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
76N06 Compressible Navier-Stokes equations
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76F65 Direct numerical and large eddy simulation of turbulence
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[1] Chapman, D., Computational aerodynamics development and outlook, AIAA J., 17, 12, 1293-1313 (1979) · Zbl 0443.76060
[2] Kuntzmann, J., Neure Entwicklungen der Methoden von Runge und Kutta, Z. Angew. Math. Phys., 41, 29-31 (1961) · Zbl 0106.10403
[3] Butcher, J., Implicit Runge-Kutta processes, Math. Comput., 18, 85, 50 (1964) · Zbl 0123.11701
[4] Gear, G., Numerical Initial Value Problems in Ordinary Differential Equations (1971) · Zbl 1145.65316
[5] Sengupta, T. K.; Ganeriwal, G.; De, S., Analysis of central and upwind compact schemes, J. Comput. Phys., 192, 677-694 (2003) · Zbl 1038.65082
[6] Sengupta, T. K.; Rajpoot, M. J.; Bhumkar, Y. G., Space-time discretizing optimal DRP schemes for flow and wave propagation problems, J. Comput. Phys., 47, 144-154 (2011) · Zbl 1271.76219
[7] Runge, C., Über die numerische auflösung von differentialgleichungen, Math. Ann., 46, 167-178 (1895) · JFM 26.0341.01
[8] Kutta, W., Beitrag zur näherungsweisen Integration totaler Differentialgleichungen, Z. Angew. Math. Phys., 46, 435-453 (1901) · JFM 32.0316.02
[9] Bogey, C.; Bailly, C., A family of low dispersive and low dissipative explicit schemes for flow and noise computations, J. Comput. Phys., 194, 1, 194-214 (2004) · Zbl 1042.76044
[10] Cockburn, B.; Shu, C., TVB Runge-Kutta local projection Discontinuous Galerkin finite method for conservation laws II: general framework, Math. Comput., 52, 186, 411-435 (1989) · Zbl 0662.65083
[11] Cockburn, B.; Lin, S.; Shu, C., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84, 90-113 (1989) · Zbl 0677.65093
[12] Cockburn, B.; Hou, S.; Shu, C., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 84, 190, 545-581 (1990) · Zbl 0695.65066
[13] Cockburn, B.; Shu, C., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 2, 199-224 (1998) · Zbl 0920.65059
[14] Gottlieb, S.; Shu, C., Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 221, 73-85 (1998) · Zbl 0897.65058
[15] Williamson, J., Low-storage Runge-Kutta schemes, J. Comput. Phys., 35, 1, 48-56 (1980) · Zbl 0425.65038
[16] Norsett, S., An A-stable modification of the Adams-Bashforth methods, (Lecture Notes in Mathematics (1969), Springer: Springer Berlin, Heidelberg), 214-219 · Zbl 0188.22502
[17] Higham, D. J.; Trefethen, L. N., Stiffness of ODEs, BIT Numer. Math., 33, 2, 285-303 (1993) · Zbl 0782.65091
[18] Catchirayer, M.; Boussuge, J-F.; Sagaut, P.; Montagnac, M.; Papadogiannis, D.; Garnaud, X., Extended integral wall-model for large-eddy simulations of compressible wall-bounded turbulent flows, Phys. Fluids, 30, 6, Article 065106 pp. (2018)
[19] Spalart, P. R., Detached-eddy simulation, Annu. Rev. Fluid Mech., 41, 1, 181-202 (2009) · Zbl 1159.76036
[20] Sagaut, P.; Deck, S.; Terracol, M., Multiscale and Multiresolution Approaches in Turbulence (2006) · Zbl 1107.76003
[21] Limare, A.; Brenner, P.; Borouchaki, H., An adaptive remeshing strategy for unsteady aerodynamics applications, (46th AIAA Fluid Dynamics Conference. 46th AIAA Fluid Dynamics Conference, Washington, D.C., June 13-17 (2016)), AIAA Paper 2016-3180
[22] Pont, G.; Brenner, P.; Cinnella, P.; Maugars, B.; Robinet, J., Multiple-correction hybrid k-exact schemes for high-order compressible RANS-LES simulations on fully unstructured grids, J. Comput. Phys., 350, 45-83 (2017) · Zbl 1380.76066
[23] Charrier, L.; Pont, G.; Marié, S.; Brenner, P.; Grasso, F., Hybrid RANS/LES simulation of a supersonic coaxial He/air jet experiment at various turbulent Lewis numbers, (Progress in Hybrid RANS-LES Modelling (2018), Springer International Publishing), 337-346
[24] Carpaye, J. C.; Roman, J.; Brenner, P., Design and analysis of a task-based parallelization over a runtime system of an explicit finite-volume CFD code with adaptive time stepping, J. Comput. Sci., 28, 439-454 (2018)
[25] Menasria, A.; Brenner, P.; Cinnella, P.; Pont, G., Toward an improved wall treatment for multiple-correction k-exact schemes, (2018 Fluid Dynamics Conference. 2018 Fluid Dynamics Conference, AIAA Aviation Forum, Atlanta, Georgia, June 25-29 (2018)), AIAA Paper 2018-4164
[26] Menasria, A.; Brenner, P.; Cinnella, P., Improving the treatment of near-wall regions for multiple-correction k-exact schemes, Comput. Fluids, 181, 116-134 (2019) · Zbl 1410.76251
[27] Heun, K., Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen veränderlichen, Z. Angew. Math. Phys., 45, 23-38 (1900) · JFM 31.0333.02
[28] Krivodonova, L., An efficient local time-stepping scheme for solution of nonlinear conservation laws, J. Comput. Phys., 229, 22, 8537-8551 (2010) · Zbl 1201.65171
[29] Strang, G., On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5, 3, 506-517 (1968) · Zbl 0184.38503
[30] Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 2, 308-323 (1985) · Zbl 0582.76038
[31] Ascher, U.; Ruuth, S.; Wetton, B., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 3, 797-823 (1995) · Zbl 0841.65081
[32] 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, 2, 1753-1781 (2007) · Zbl 1123.65097
[33] Vermeire, B. C.; Nadarajah, S., Adaptive IMEX schemes for high-order unstructured methods, J. Comput. Phys., 280, 261-286 (2015) · Zbl 1349.76545
[34] Bertolazzi, E.; Manzini, G., DIMEX Runge-Kutta finite volume methods for multidimensional hyperbolic systems, Math. Comput. Simul., 75, 5, 141-160 (2007) · Zbl 1124.65073
[35] Manzini, G., A second-order TVD implicit-explicit finite volume method for time-dependent convection-reaction equations, Math. Comput. Simul., 79, 8, 2403-2428 (2009) · Zbl 1180.65110
[36] Fryxell, B. A.; Woodward, P. R.; Colella, P.; Winkler, K.-H., An implicit-explicit hybrid method for Lagrangian hydrodynamics, J. Comput. Phys., 63, 2, 283-310 (1986) · Zbl 0596.76078
[37] Dai, W.; Woodward, P. R., A second-order iterative implicit-explicit hybrid scheme for hyperbolic systems of conservation laws, J. Comput. Phys., 128, 1, 181-196 (1996) · Zbl 0863.65050
[38] Collins, J.; Colella, P.; Glaz, H., An implicit-explicit Eulerian Godunov scheme for compressible flow, J. Comput. Phys., 116, 2, 195-211 (1995) · Zbl 0817.76038
[39] Men’shov, I.; Nakamura, Y., Hybrid explicit-implicit, unconditionally stable scheme for unsteady compressible flows, AIAA J., 42, 3, 551-559 (2004)
[40] Timofeev, E.; Norouzi, F., Hybrid, explicit-implicit, finite-volume schemes on unstructured grids for unsteady compressible flows, (AIP Conference Proceedings, vol. 1738 (2016)), 030002
[41] Tóth, G.; De Zeeuw, D.; Gombosi, T.; Powell, K., A parallel explicit/implicit time stepping scheme on block-adaptive grids, J. Comput. Phys., 217, 2, 722-758 (2006) · Zbl 1178.76287
[42] May, S.; Berger, M., An explicit implicit scheme for cut cells in embedded boundary meshes, J. Sci. Comput., 71, 3, 919-943 (2017) · Zbl 1372.65250
[43] P.-O. Persson, High-order LES simulations using implicit-explicit Runge-Kutta schemes, in: 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2011, https://doi.org/10.2514/6.2011-684; P.-O. Persson, High-order LES simulations using implicit-explicit Runge-Kutta schemes, in: 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, 2011, https://doi.org/10.2514/6.2011-684
[44] Haider, F.; Brenner, P.; Courbet, B.; Croisille, J., Parallel implementation of k-exact finite volume reconstruction on unstructured grids, (Lecture Notes in Computational Science and Engineering (2014), Springer International Publishing), 59-75 · Zbl 1426.76374
[45] Ollivier-Gooch, C.; Van Altena, M., A high-order-accurate unstructured mesh finite-volume scheme for the advection – diffusion equation, J. Comput. Phys., 181, 2, 729-752 (2002) · Zbl 1178.76251
[46] Crank, J.; Nicolson, P., A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Camb. Philos. Soc., 43, 50-67 (1947) · Zbl 0029.05901
[47] Norouzi, F.; Timofeev, E., A hybrid, explicit-implicit, second order in space and time TVD scheme for one-dimensional scalar conservation laws, (20th AIAA CFD Conference. 20th AIAA CFD Conference, Honolulu, Hawaii, June 27-30 (2011)), AIAA Paper 2011-3046
[48] Timofeev, E.; Norouzi, F., Application of a new hybrid explicit-implicit flow solver to 1d unsteady flows with shock waves, (Kontis, K., Proc. of the 28th ISSW, vol. 2. Proc. of the 28th ISSW, vol. 2, Manchester, UK, July 17-22 (2011))
[49] Norouzi, F.; Timofeev, E., A hybrid, explicit-implicit, second order in space and time TVD scheme for two-dimensional compressible flows, (E. Oñate; etal., Proc. of the WCCM XI-ECCM V-ECCM VI, vol. 2. Proc. of the WCCM XI-ECCM V-ECCM VI, vol. 2, Barcelona, Spain, July 20-25 (2014)), 4820-4831
[50] Sengupta, T. K.; Bhumkar, Y. G.; Rajpoot, M. K.; Suman, V.; Saurabh, S., Spurious waves in discrete computation of wave phenomena and flow problems, Appl. Math. Comput., 218, 18, 9035-9065 (2012) · Zbl 1245.65112
[51] Vanharen, J.; Puigt, G.; Vasseur, X.; Boussuge, J-F.; Sagaut, P., Revisiting the spectral analysis for high-order spectral discontinuous methods, J. Comput. Phys., 337, 379-402 (2017) · Zbl 1415.76577
[52] Vichnevetsky, R., Energy and group velocity in semi discretizations of hyperbolic equations, Math. Comput. Simul., 23, 333-343 (1981) · Zbl 0524.65061
[53] Vichnevetsky, R.; Bowles, J. B., Fourier Analysis of Numerical Approximations of Hyperbolic Equations (1982), Society for Industrial and Applied Mathematics · Zbl 0495.65041
[54] Poinsot, T.; Veynante, D., Theoretical and Numerical Combustion (2005), R.T. Edwards Inc.
[55] Trefethen, L. N., Group velocity in finite difference schemes, SIAM Rev., 24, 2, 113-136 (1992) · Zbl 0487.65055
[56] Cinnella, P.; Content, C., High-order implicit residual smoothing time scheme for direct and large eddy simulations of compressible flows, J. Comput. Phys., 326, 1-29 (2016) · Zbl 1373.76058
[57] Martín, M. P.; Candler, G. V., A parallel implicit method for the direct numerical simulation of wall-bounded compressible turbulence, J. Comput. Phys., 215, 1, 153-171 (2006) · Zbl 1088.76021
[58] Roe, P. L., Characteristic-based schemes for the Euler equations, Annu. Rev. Fluid Mech., 18, 1, 337-365 (1986) · Zbl 0624.76093
[59] Pont, G., Self Adaptive Turbulence Models for Unsteady Compressible Flows (2015), ENAM, Master’s thesis
[60] Limare, A., Adaptation par enrichissement de maillages intersectants, dans un contexte Volume Finis d’ordre élévé, pour la simulation des écoulements compressible instationnaires (2017), Université de Technologie de Troyes, Ph.D. thesis
[61] Hall, S.; Behnia, M.; Fletcher, C.; Morrison, G., Investigation of the secondary corner vortex in a benchmark turbulent backward-facing step using cross-correlation particle imaging velocimetry, Exp. Fluids, 35, 2, 139-151 (2003)
[62] Spalart, P.; Deck, S.; Shur, M.; Squires, K.; Strelets, M.; Travin, A., A new version of detached-eddy simulation, resistant to ambiguous grid densities, Theor. Comput. Fluid Dyn., 20, 181-195 (2006) · Zbl 1112.76370
[63] Deck, S., Recent improvements in the Zonal Detached Eddy Simulation (ZDES) formulation, Theor. Comput. Fluid Dyn., 26, 6, 523-550 (2012)
[64] Spalart, P.; Allmaras, S., A one-equation turbulence model for aerodynamic flows, AIAA, 439 (1992)
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