A study of several artificial viscosity models within the discontinuous Galerkin framework. (English) Zbl 1491.76046
Summary: Dealing with strong shocks while retaining low numerical dissipation traditionally has been one of the major challenges for high order methods like discontinuous Galerkin (DG). In the literature, shock capturing models have been designed for DG based on various approaches, such as slope limiting, (H)WENO reconstruction, a posteriori sub-cell limiting, and artificial viscosity, among which a subclass of artificial viscosity methods are compared in the present work. Four models are evaluated, including a dilation-based model, a highest modal decay model, an averaged modal decay model, and an entropy viscosity model. Performance for smooth, non-smooth and broadband problems are examined with typical one- and two-dimensional cases.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76N15 | Gas dynamics (general theory) |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
high order discontinuous Galerkin method; shock capturing; dilation-based model; highest modal decay model; averaged modal decay model; entropy viscosity modelReferences:
| [1] | N. Kroll, ADIGMA: A European project on the development of adaptive higher order vari-ational methods for aerospace applications, in: 47th AIAA Aerospace Sciences Meeting in-cluding The New Horizons Forum and Aerospace Exposition, 2009, p. 176. |
| [2] | Z. J. Wang, K. Fidkowski, R. Abgrall, F. Bassi, D. Caraeni, A. Cary, H. Deconinck, R. Hart-mann, K. Hillewaert, H. T. Huynh, et al., High-order CFD methods: current status and perspective, International Journal for Numerical Methods in Fluids 72 (8) (2013) 811-845. · Zbl 1455.76007 |
| [3] | J. S. Hesthaven, T. Warburton, Nodal discontinuous Galerkin methods: algorithms, analysis, and applications, Springer Science & Business Media, 2007. · Zbl 1134.65068 |
| [4] | B. Cockburn, S.-Y. Lin, C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, Journal of Com-putational Physics 84 (1) (1989) 90-113. · Zbl 0677.65093 |
| [5] | B. Cockburn, C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conser-vation laws V: Multidimensional systems, Journal of Computational Physics 141 (2) (1998) 199-224. · Zbl 0920.65059 |
| [6] | A. Burbeau, P. Sagaut, C.-H. Bruneau, A problem-independent limiter for high-order Runge-Kutta discontinuous Galerkin methods, Journal of Computational Physics 169 (1) (2001) 111-150. · Zbl 0979.65081 |
| [7] | L. Krivodonova, Limiters for high-order discontinuous Galerkin methods, Journal of Com-putational Physics 226 (1) (2007) 879-896. · Zbl 1125.65091 |
| [8] | D. Kuzmin, Hierarchical slope limiting in explicit and implicit discontinuous galerkin meth-ods, Journal of Computational Physics 257 (2014) 1140-1162. · Zbl 1352.65355 |
| [9] | J. Qiu, C.-W. Shu, Runge-Kutta discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (3) (2005) 907-929. · Zbl 1077.65109 |
| [10] | J.Zhu, J.Qiu, C.-W.Shu, M.Dumbser, Runge-Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes, Journal of Computational Physics 227 (9) (2008) 4330-4353. · Zbl 1157.65453 |
| [11] | J.Qiu, C.-W.Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: One-dimensional case, Journal of Computational Physics 193 (1) (2004) 115-135. · Zbl 1039.65068 |
| [12] | J.Qiu, C.-W.Shu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case, Computers & Fluids 34 (6) (2005) 642-663. · Zbl 1134.65358 |
| [13] | J. Zhu, J. Qiu, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method, III: Unstructured meshes, Journal of Scientific Computing 39 (2) (2009) 293-321. · Zbl 1203.65156 |
| [14] | H. Luo, J. D. Baum, R. Lohner, A Hermite WENO-based limiter for discontinuous Galerkin method on unstructured grids, Journal of Computational Physics 225 (1) (2007) 686-713. · Zbl 1122.65089 |
| [15] | X. Zhong, C.-W. Shu, A simple weighted essentially non-oscillatory limiter for Runge-Kutta discontinuous Galerkin methods, Journal of Computational Physics 232 (1) (2013) 397-415. |
| [16] | J. Zhu, X. Zhong, C.-W. Shu, J. Qiu, Runge-Kutta discontinuous Galerkin method using a new type of WENO limiters on unstructured meshes, Journal of Computational Physics 248 (2013) 200-220. · Zbl 1349.65501 |
| [17] | M. Dumbser, O. Zanotti, R. Loube’re, S. Diot, A posteriori subcell limiting of the discontin-uous galerkin finite element method for hyperbolic conservation laws, Journal of Computa-tional Physics 278 (2014) 47-75. · Zbl 1349.65448 |
| [18] | M. Dumbser, R. Loube’re, A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous galerkin method on unstructured meshes, Journal of Compu-tational Physics 319 (2016) 163-199. · Zbl 1349.65447 |
| [19] | S.Clain, S.Diot, R.Loube’re, A high-order finite volume method for systems of conservation lawsmulti-dimensional optimal order detection (mood), Journal of Computational Physics 230 (10) (2011) 4028-4050. · Zbl 1218.65091 |
| [20] | R. Loubere, M. Dumbser, S. Diot, A new family of high order unstructured mood and ader finite volume schemes for multidimensional systems of hyperbolic conservation laws, Com-munications in Computational Physics 16 (3) (2014) 718-763. · Zbl 1373.76137 |
| [21] | F. Vilar, A posteriori correction of high-order discontinuous galerkin scheme through subcell finite volume formulation and flux reconstruction, Journal of Computational Physics 387 (2019) 245-279. · Zbl 1452.65251 |
| [22] | M. Yang, Z.-J. Wang, A parameter-free generalized moment limiter for high-order methods on unstructured grids, Advances in Applied Mathematics and Mechanics 1 (2009) 451-480. |
| [23] | J. S. Park, C. Kim, Hierarchical multi-dimensional limiting strategy for correction pro-ce-dure via reconstruction, Journal of Computational Physics 308 (2016) 57-80. · Zbl 1351.76076 |
| [24] | J. S. Park, S.-H. Yoon, C. Kim, Multi-dimensional limiting process for hyperbolic con-serva-tion laws on unstructured grids, Journal of Computational Physics 229 (3) (2010) 788-812. · Zbl 1185.65150 |
| [25] | D. Kuzmin, Slope limiting for discontinuous galerkin approximations with a possibly non-orthogonal taylor basis, International Journal for Numerical Methods in Fluids 71 (9) (2013) 1178-1190. · Zbl 1431.65173 |
| [26] | L. Li, Q. Zhang, A new vertex-based limiting approach for nodal discontinuous galerkin methods on arbitrary unstructured meshes, Computers & Fluids 159 (2017) 316-326. · Zbl 1390.76331 |
| [27] | A. W. Cook, W. H. Cabot, A high-wavenumber viscosity for high-resolution numerical meth-ods, Journal of Computational Physics 195 (2) (2004) 594-601. · Zbl 1115.76366 |
| [28] | A. W. Cook, W. H. Cabot, Hyperviscosity for shock-turbulence interactions, Journal of Com-putational Physics 203 (2) (2005) 379-385. · Zbl 1143.76477 |
| [29] | B. Fiorina, S. K. Lele, An artificial nonlinear diffusivity method for supersonic reacting flows with shocks, Journal of Computational Physics 222 (1) (2007) 246-264. · Zbl 1216.76030 |
| [30] | S. Kawai, S. K. Lele, Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes, Journal of Computational Physics 227 (22) (2008) 9498-9526. · Zbl 1359.76223 |
| [31] | A. Mani, J. Larsson, P. Moin, Suitability of artificial bulk viscosity for large-eddy simu-lation of turbulent flows with shocks, Journal of Computational Physics 228 (19) (2009) 7368-7374. · Zbl 1421.76133 |
| [32] | S. Premasuthan, C. Liang, A. Jameson, Computation of flows with shocks using the spectral difference method with artificial viscosity, I: Basic formulation and application, Computers & Fluids 98 (2014) 111-121. · Zbl 1391.76488 |
| [33] | D.Moro, N.C.Nguyen, J.Peraire, Dilation-based shock capturing for high-order methods, International Journal for Numerical Methods in Fluids 82 (7) (2016) 398-416. |
| [34] | P.-O.Persson, J.Peraire,Sub-cell shock capturing for discontinuous Galerkin methods, in: 44th AIAA Aerospace Sciences Meeting and Exhibit, AIAA paper 2006-112, 2006. |
| [35] | A. Klockner, T. Warburton, J. S. Hesthaven, Viscous shock capturing in a time-explicit dis-continuous Galerkin method, Mathematical Modelling of Natural Phenomena 6 (3) (2011) 57-83. · Zbl 1220.65165 |
| [36] | J.-L.Guermond, R.Pasquetti, B.Popov, Entropy viscosity method for nonlinear conservation laws, Journal of Computational Physics 230 (11) (2011) 4248-4267. · Zbl 1220.65134 |
| [37] | V. Zingan, J.-L. Guermond, J. Morel, B. Popov, Implementation of the entropy viscos-ity method with the discontinuous Galerkin method, Computer Methods in Applied Mechanics and Engineering 253 (2013) 479-490. · Zbl 1297.76109 |
| [38] | A.Chaudhuri, G.B.Jacobs, W.-S.Don, H.Abbassi, F.Mashayek, Explicit discontinuous spec-tral element method with entropy generation based artificial viscosity for shocked viscous flows, Journal of Computational Physics 332 (2017) 99-117. · Zbl 1378.76040 |
| [39] | R. Hartmann, Adaptive discontinuous Galerkin methods with shock-capturing for the com-pressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 51 (9-10) (2006) 1131-1156. · Zbl 1106.76041 |
| [40] | F. Bassi, A. Crivellini, A. Ghidoni, S. Rebay, High-order discontinuous Galerkin dis-cretiza-tion of transonic turbulent flows, in: 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, AIAA paper 2009-180, 2009. |
| [41] | P. Fernandez, N.-C. Nguyen, J. Peraire, A physics-based shock capturing method for large-eddy simulation, arXiv preprint arXiv:1806.06449 (2018). |
| [42] | T. Haga, S. Kawai, On a robust and accurate localized artificial diffusivity scheme for the high-order flux-reconstruction method, Journal of Computational Physics 376 (2019) 534-563. · Zbl 1416.76110 |
| [43] | S. S. Collis, K. Ghayour, Discontinuous galerkin methods for compressible dns, in: ASME/JSME 2003 4th Joint Fluids Summer Engineering Conference, American Soci-ety of Mechanical Engineers, 2003, pp. 1777-1786. |
| [44] | L.Wei, A.Pollard, Direct numerical simulation of compressible turbulent channel flows using the discontinuous galerkin method, Computers & Fluids 47 (1) (2011) 85-100. · Zbl 1271.76142 |
| [45] | J.-B.Chapelier, M.DeLaLlavePlata, F.Renac, E.Lamballais, Evaluation of a high-order discon-tinuous Galerkin method for the dns of turbulent flows, Computers & Fluids 95 (2014) 210-226. · Zbl 1391.76209 |
| [46] | A. D. Beck, T. Bolemann, D. Flad, H. Frank, G. J. Gassner, F. Hindenlang, C.-D. Munz, High-order discontinuous galerkin spectral element methods for transitional and tur-bulent flow simulations, International Journal for Numerical Methods in Fluids 76 (8) (2014) 522-548. |
| [47] | J. Yu, C. Yan, Z. Jiang, Suitability of artificial viscosity discontinuous galerkin method for compressible turbulence, Science China Technological Sciences 60 (7) (2017) 1032-1049. |
| [48] | J.Yu, C.Yan, Z.Jiang, Effects of artificial viscosity and upwinding on spectral properties of the discontinuous galerkin method, Computers & Fluids 175 (2018) 276-292. · Zbl 1410.76074 |
| [49] | G. E. Barter, D. L. Darmofal, Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation, Journal of Computational Physics 229 (5) (2010) 1810-1827. · Zbl 1329.76153 |
| [50] | H. Abbassi, F. Mashayek, G. B. Jacobs, Shock capturing with entropy-based artificial viscos-ity for staggered grid discontinuous spectral element method, Computers & Fluids 98 (2014) 152-163. · Zbl 1391.76506 |
| [51] | M. H. Carpenter, C. A. Kennedy, Fourth-order 2N-storage Runge-Kutta schemes, NASA Report TM 109112, NASA Langley Research Center (1994). |
| [52] | J. S. Hesthaven, Numerical methods for conservation laws: From analysis to algo-rithms, SIAM Publishing, 2017. |
| [53] | R. C. Moura, S. Sherwin, J. Peiro, Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous galerkin spectral/hp methods, Journal of Computational Physics 298 (2015) 695-710. · Zbl 1349.76131 |
| [54] | A. Bahraminasab, M. Niry, J. Davoudi, M. R. R. Tabar, A. Masoudi, K. Sreenivasan, Taylors frozen-flow hypothesis in burgers turbulence, Physical Review E 77 (6) (2008) 065302. |
| [55] | A. Campos, B. Morgan, The effect of artificial bulk viscosity in simulations of forced com-pressible turbulence, Journal of Computational Physics 371 (2018) 111-121. · Zbl 1415.76391 |
| [56] | 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, p. 684. |
| [57] | C. C. de Wiart, K. Hillewaert, Dns and iles of transitional flows around a sd7003 using a high order discontinuous galerkin method, in: Seventh international conference on compu-tational fluid dynamics (ICCFD7), Big Island, Hawaii, 2012. |
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