Hybridizable discontinuous Galerkin projection methods for Navier-Stokes and Boussinesq equations. (English) Zbl 1351.76083
Summary: Schemes for the incompressible Navier-Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit-Explicit Runge-Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed-element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
hybridizable DG; projection methods; IMEX-RK; high-order; selective limiter; quadrature-free; Navier-Stokes; Boussinesq; ocean modelingReferences:
| [1] | Ahnert, T.; Bärwolff, G., Numerical comparison of hybridized discontinuous Galerkin and finite volume methods for incompressible flow, Int. J. Numer. Methods Fluids, 76, 267-281 (2014) · Zbl 1455.76032 |
| [2] | Arnold, D. N.; Brezzi, F.; Cockburn, B.; Marini, L. D., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39, 5, 1749-1779 (2002) · Zbl 1008.65080 |
| [3] | Ascher, U.; Ruuth, S.; Spiteri, R., Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Appl. Numer. Math., 25, 151-167 (1997) · Zbl 0896.65061 |
| [4] | Barter, G. E.; Darmofal, D. L., Shock capturing with higher-order, PDE-based artificial viscosity (2007), AIAA paper 3823:2007 |
| [5] | Barter, G. E.; Darmofal, D. L., Shock capturing with PDE-based artificial viscosity for DGFEM: Part i. formulation, J. Comp. Physiol., 229, 5, 1810-1827 (2010) · Zbl 1329.76153 |
| [6] | Blossey, P. N.; Durran, D. R., Selective monotonicity preservation in scalar advection, J. Comput. Phys., 227, 10, 5160-5183 (2008) · Zbl 1142.65069 |
| [7] | Chorin, A., Numerical solution of the Navier-Stokes equations, Math. Comput., 22, 745-762 (1968) · Zbl 0198.50103 |
| [8] | Cockburn, B.; Gopalakrishnan, J., The derivation of hybridizable discontinuous Galerkin methods for Stokes flow, SIAM J. Numer. Anal., 47, 2, 1092-1125 (2009) · Zbl 1279.76016 |
| [9] | Cockburn, B.; Shu, C. W., Tvb Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework, Math. Comput., 52, 186, 411-435 (1989) · Zbl 0662.65083 |
| [10] | Cockburn, B.; Shu, C. W., The Runge-Kutta discontinuous Galerkin method for conservation laws. V: multidimensional systems, J. Comput. Phys., 141, 2, 199-224 (1998) · Zbl 0920.65059 |
| [11] | Cockburn, B.; Shu, C. W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 3, 173-261 (2001) · Zbl 1065.76135 |
| [12] | Cockburn, B.; Lin, S. Y.; Shu, C. W., Tvb Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84, 1, 90-113 (1989) · Zbl 0677.65093 |
| [13] | Cockburn, B.; Hou, S.; Shu, C. W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. iv. the multidimensional case, Math. Comput., 54, 190, 545-581 (1990) · Zbl 0695.65066 |
| [14] | Cockburn, B.; Gopalakrishnan, J.; Lazarov, R., Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 2, 1319-1365 (2009) · Zbl 1205.65312 |
| [15] | Cockburn, B.; Guzmán, J.; Wang, H., Superconvergent discontinuous Galerkin methods for second-order elliptic problems, Math. Comput., 78, 265, 1-24 (2009) · Zbl 1198.65194 |
| [16] | Cockburn, B.; Gopalakrishnan, J.; Nguyen, N.; Peraire, J.; Sayas, F. J., Analysis of HDG methods for Stokes flow, Math. Comput., 80, 274, 723-760 (2011) · Zbl 1410.76164 |
| [17] | Deleersnijder, E.; Legat, V.; Lermusiaux, P. F.J., Multi-scale modelling of coastal, shelf and global ocean dynamics, Ocean Dyn., 60, 6, 1357-1359 (2010) |
| [18] | Denaro, F. M., On the application of the Helmholtz-Hodge decomposition in projection methods for incompressible flows with general boundary conditions, Int. J. Numer. Methods Fluids, 43, 1, 43-69 (2003) · Zbl 1032.76636 |
| [19] | Durran, D. R., Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Texts Appl. Math., vol. 32 (1999), Springer-Verlag |
| [20] | Ferziger, J. H.; Peric, M., Computational Methods for Fluid Dynamics (2002), Springer: Springer New York, NY · Zbl 0998.76001 |
| [21] | Fringer, O.; Gerritsen, M.; Street, R., An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator, Ocean Model., 14, 3, 139-173 (2006) |
| [22] | Geiser, J., Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations, J. Comput. Appl. Math., 217, 1, 227-242 (2008) · Zbl 1144.65062 |
| [23] | Guermond, J.; Minev, P.; Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195, 6011-6045 (2006) · Zbl 1122.76072 |
| [24] | Haley, P. J.; Lermusiaux, P. F.J., Multiscale two-way embedding schemes for free-surface primitive equations in the “Multidisciplinary Simulation, Estimation and Assimilation System”, Ocean Dyn., 60, 6, 1497-1537 (2010) |
| [25] | Härtel, C.; Meiburg, E.; Necker, F., Analysis and direct numerical simulation of the flow at a gravity-current head. Part 1. Flow topology and front speed for slip and no-slip boundaries, J. Fluid Mech., 418, 189-212 (2000) · Zbl 0985.76042 |
| [26] | Hesthaven, J.; Kirby, R., Filtering in Legendre spectral methods, Math. Comput., 77, 263, 1425-1452 (2008) · Zbl 1195.65138 |
| [27] | Hesthaven, J.; Warburton, T., Nodal Discontinuous Galerkin Methods, Texts Appl. Math., vol. 54 (2008), Springer: Springer New York, NY · Zbl 1134.65068 |
| [28] | Hoteit, H.; Ackerer, P.; Mosé, R.; Erhel, J.; Philippe, B., New two-dimensional slope limiters for discontinuous Galerkin methods on arbitrary meshes, Int. J. Numer. Methods Eng., 61, 14, 2566-2593 (2004) · Zbl 1075.76575 |
| [29] | Huerta, A.; Casoni, E.; Peraire, J., A simple shock-capturing technique for high-order discontinuous Galerkin methods, Int. J. Numer. Methods Fluids, 69, 10, 1614-1632 (2012) · Zbl 1253.76058 |
| [30] | Huynh, L.; Nguyen, N.; Peraire, J.; Khoo, B., A high-order hybridizable discontinuous Galerkin method for elliptic interface problems, Int. J. Numer. Methods Eng., 93, 2, 183-200 (2013) · Zbl 1352.65513 |
| [31] | Kennedy, C.; Carpenter, M., Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Appl. Numer. Math., 44, 139-181 (2003) · Zbl 1013.65103 |
| [32] | Kirby, R. M.; Sherwin, S. J.; Cockburn, B., To cg or to hdg: a comparative study, J. Sci. Comput., 51, 1, 183-212 (2012) · Zbl 1244.65174 |
| [33] | Krivodonova, L., Limiters for high-order discontinuous Galerkin methods, J. Comput. Phys., 226, 1, 276-296 (2007) · Zbl 1125.65091 |
| [34] | Lermusiaux, P. F.J.; Schröter, J.; Danilov, S.; Iskandarani, M.; Pinardi, N.; Westerink, J. J., Multiscale modeling of coastal, shelf and global ocean dynamics, Ocean Dyn., 63, 11-12, 1341-1344 (2013) |
| [35] | Lermusiaux, P. F.J.; Ueckermann, M. P.; Mirabito, C.; Haley, P. J.; Aoussou, J., High order hybridizable discontinuous Galerkin projection schemes for incompressible Navier-Stokes and ocean primitive equations: derivation details (2014), Department of Mechanical Engineering, Massachusetts Institute of Technology: Department of Mechanical Engineering, Massachusetts Institute of Technology Cambridge, Massachusetts, USA, MSEAS Report 20 |
| [36] | Mavriplis, C. A., Nonconforming discretization and a posteriori error estimators for adaptive spectral element techniques (1989), MIT, Ph.D. Thesis |
| [37] | Michoski, C.; Mirabito, C.; Dawson, C.; Wirasaet, D.; Kubatko, E.; Westerink, J., Adaptive hierarchic transformations for dynamically p-enriched slope-limiting over discontinuous Galerkin systems of generalized equations, J. Comput. Phys., 230, 22, 8028-8056 (2011) · Zbl 1269.65099 |
| [39] | Nguyen, N.; Persson, P. O.; Peraire, J., Rans solutions using high order discontinuous Galerkin methods (2007), AIAA Paper 914 |
| [40] | Nguyen, N.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations, J. Comput. Phys., 228, 9, 3232-3254 (2009) · Zbl 1187.65110 |
| [41] | Nguyen, N.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for nonlinear convection-diffusion equations, J. Comput. Phys., 228, 23, 8841-8855 (2009) · Zbl 1177.65150 |
| [42] | Nguyen, N.; Peraire, J.; Cockburn, B., A hybridizable discontinuous Galerkin method for Stokes flow, Comput. Methods Appl. Mech. Eng., 199, 9, 582-597 (2010) · Zbl 1227.76036 |
| [43] | Nguyen, N.; Peraire, J.; Cockburn, B., A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, (Proc. of the 48th AIAA Aerospace Sciences Meeting and Exhibit (2010)), Orlando, FL · Zbl 1227.76036 |
| [44] | Nguyen, N.; Peraire, J.; Cockburn, B., An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 230, 4, 1147-1170 (2011) · Zbl 1391.76353 |
| [45] | Nguyen, N.; Roca, X.; Moro, D.; Peraire, J., A hybridized multiscale discontinuous Galerkin method for compressible flows (2013), AIAA Paper 689 |
| [46] | Oreskes, N.; Shrader-Frechette, K.; Belitz, K., Verification, validation, and confirmation of numerical models in the earth sciences, Science, 263, 5147, 641-646 (1994) |
| [47] | Solano, M. E., Hybridizable discontinuous Galerkin method for curved domains (2012), University of Minnesota, PhD thesis |
| [48] | Peraire, J.; Nguyen, N.; Cockburn, B., A hybridizable discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations (2010), AIAA Paper 363:2010 · Zbl 1227.76036 |
| [49] | Peraire, J.; Nguyen, N.; Cockburn, B., An embedded discontinuous Galerkin method for the compressible Euler and Navier-Stokes equations (2011), AIAA Paper 3228 · Zbl 1391.76353 |
| [50] | Persson, P. O.; Peraire, J., Sub-cell shock capturing for discontinuous Galerkin methods (2006), AIAA paper 112 |
| [51] | Qiu, J.; Shu, C. W., Runge-Kutta discontinuous Galerkin method using weno limiters, SIAM J. Sci. Comput., 26, 3, 907-929 (2005) · Zbl 1077.65109 |
| [52] | Roache, P. J., Verification and Validation in Computational Science and Engineering (1998), Hermosa: Hermosa Albuquerque |
| [53] | Schütz, J.; May, G., A hybrid mixed method for the compressible Navier-Stokes equations, J. Comput. Phys., 240, 58-75 (2013) · Zbl 1426.76566 |
| [54] | Strang, G.; Fix, G. J., An Analysis of the Finite Element Method, vol. 212 (1973), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0278.65116 |
| [55] | Témam, R., Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II), Arch. Ration. Mech. Anal., 33, 377-385 (1969) · Zbl 0207.16904 |
| [56] | Timmermans, L.; Minev, P.; Van De Vosse, F., An approximate projection scheme for incompressible flow using spectral elements, Int. J. Numer. Methods Fluids, 22, 673-688 (1996) · Zbl 0865.76070 |
| [57] | Ueckermann, M., Towards next generation ocean models: novel discontinuous Galerkin schemes for 2D unsteady biogeochemical models (2009), Massachusetts Institute of Technology, Department of Mechanical Engineering, Master’s thesis |
| [58] | Ueckermann, M., High order hybrid discontinuous Galerkin regional ocean modelling (2014), Massachusetts Institute of Technology, Department of Mechanical Engineering, PhD thesis |
| [59] | Ueckermann, M. P.; Lermusiaux, P. F.J., High-order schemes for 2D unsteady biogeochemical ocean models, Ocean Dyn., 60, 1415-1445 (2010) |
| [61] | Vincent, P.; 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 |
| [62] | Waluga, C.; Egger, H., An implementation of hybrid discontinuous Galerkin methods in dune, (Advances in DUNE (2012), Springer), 169-180 |
| [63] | Zhu, J.; Qiu, J.; Shu, C. W.; Dumbser, M., Runge-Kutta discontinuous Galerkin method using weno limiters II: unstructured meshes, J. Comput. Phys., 227, 9, 4330-4353 (2008) · Zbl 1157.65453 |
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.
