A two-stage fourth-order gas-kinetic CPR method for the Navier-Stokes equations on triangular meshes. (English) Zbl 07517149
Summary: An efficient gas-kinetic scheme with fourth-order accuracy in both space and time is developed for the Navier-Stokes equations on triangular meshes. The scheme combines an efficient correction procedure via reconstruction (CPR) framework with a robust gas-kinetic flux formula, which computes both the flux and its time-derivative. The availability of the flux and its time-derivative makes it straightforward to adopt an efficient two-stage temporal discretization to achieve fourth-order time accuracy. In addition, through the gas-kinetic evolution model, the inviscid and viscous fluxes are coupled and computed uniformly without any separate treatment for the viscous fluxes. As a result, the current scheme is more efficient than traditional explicit CPR methods with a separate treatment for viscous fluxes, and a fourth-order Runge-Kutta approach. Furthermore, a robust and accurate subcell finite volume limiting procedure is extended to the CPR framework for troubled cells, resulting in subcell resolution of flow discontinuities. Numerical tests demonstrate the high accuracy, efficiency and robustness of the current scheme in a wide range of inviscid and viscous flow problems from subsonic to supersonic speeds.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 76Nxx | Compressible fluids and gas dynamics |
Keywords:
gas-kinetic scheme; correction procedure via reconstruction; two-stage fourth-order temporal discretization; subcell finite volume method; high-order accurate schemeSoftware:
HE-E1GODFReferences:
| [1] | Wang, Z. J.; Fidkowski, K.; Abgrall, R.; Bassi, F.; Caraeni, D.; Cary, A.; Deconinck, H.; Hartmann, R.; Hillewaert, K.; Huynh, H.; Kroll, N.; May, G.; Persson, P.-O.; Leer, B.; Visbal, M., High-order CFD methods: current status and perspective, Int. J. Numer. Methods Fluids, 72, 8, 811-845 (2013) · Zbl 1455.76007 |
| [2] | Barth, T. J.; Frederickson, P. O., Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction (1990), AIAA Paper 1990-0013 |
| [3] | Hu, C.; Shu, C.-W., Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127 (1999) · Zbl 0926.65090 |
| [4] | Wang, Q.; Ren, Y.-X.; Pan, J.; Li, W., Compact high order finite volume method on unstructured grids III: variational reconstruction, J. Comput. Phys., 337, 1-26 (2017) · Zbl 1415.76482 |
| [5] | Zhang, Y.; Ren, Y.-X.; Wang, Q., Compact high order finite volume method on unstructured grids IV: explicit multi-step reconstruction schemes on compact stencil, J. Comput. Phys., 396, 161-192 (2019) · Zbl 1452.76141 |
| [6] | Pan, J.; Ren, Y.-X.; Sun, Y., High order sub-cell finite volume schemes for solving hyperbolic conservation laws II: extension to two-dimensional systems on unstructured grids, J. Comput. Phys., 338, 165-198 (2017) · Zbl 1415.65210 |
| [7] | Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224 (1998) · Zbl 0920.65059 |
| [8] | Luo, H.; Baum, J. D.; Löhner, R., A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 227, 8875-8893 (2008) · Zbl 1391.76350 |
| [9] | Zhang, X.; Shu, C.-W., On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, J. Comput. Phys., 229, 8918-8934 (2010) · Zbl 1282.76128 |
| [10] | Huynh, H. T., A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods (2007), AIAA Paper 2007-4079 |
| [11] | Huynh, H. T., A reconstruction approach to high-order schemes including discontinuous Galerkin for diffusion (2009), AIAA Paper 2009-403 |
| [12] | Liu, Y.; Vinokur, M.; Wang, Z. J., Spectral difference method for unstructured grids I: basic formulation, J. Comput. Phys., 216, 2, 780-801 (2006) · Zbl 1097.65089 |
| [13] | Liang, C.; Jameson, A.; Wang, Z. J., Spectral difference method for compressible flow on unstructured grids with mixed elements, J. Comput. Phys., 228, 2847-2858 (2009) · Zbl 1159.76029 |
| [14] | Wang, Z. J., Spectral (finite) volume method for conservation laws on unstructured grids. Basic formulation, J. Comput. Phys., 178, 1, 210-251 (2002) · Zbl 0997.65115 |
| [15] | Wang, Z. J.; Zhang, L.; Liu, Y., Spectral (finite) volume method for conservation laws on unstructured grids IV: extension to two-dimensional systems, J. Comput. Phys., 194, 2, 716-741 (2004) · Zbl 1039.65072 |
| [16] | Wang, Z. J.; Gao, H., A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys., 228, 8161-8186 (2009) · Zbl 1173.65343 |
| [17] | Gao, H.; Wang, Z. J., A high-order lifting collocation penalty formulation for the Navier-Stokes equations on 2-D mixed grids (2009), AIAA Paper 2009-3784 |
| [18] | Haga, T.; Gao, H.; Wang, Z. J., A high-order unifying discontinuous formulation for 3-D mixed grids (2010), AIAA Paper 2010-540 |
| [19] | Haga, T.; Gao, H.; Wang, Z. J., A high-order unifying discontinuous formulation for the Navier-Stokes equations on 3D mixed grids, Math. Model. Nat. Phenom., 6, 28-56 (2011) · Zbl 1239.76044 |
| [20] | Huynh, H. T.; Wang, Z. J.; Vincent, P. E., High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids, Comput. Fluids, 98, 209-220 (2014) · Zbl 1390.65123 |
| [21] | Park, J. S.; Kim, C., Multi-dimensional limiting process for finite volume methods on unstructured grids, Comput. Fluids, 65, 8-24 (2012) · Zbl 1365.76167 |
| [22] | Park, J. S.; Kim, C., Hierarchical multi-dimensional limiting strategy for correction procedure via reconstruction, J. Comput. Phys., 308, 57-80 (2016) · Zbl 1351.76076 |
| [23] | You, H.; Kim, C., High-order multi-dimensional limiting strategy with subcell resolution I. Two-dimensional mixed meshes, J. Comput. Phys., 375, 1005-1032 (2018) · Zbl 1416.65367 |
| [24] | Fernandez, P.; Nguyen, C.; Peraire, J., A physics-based shock capturing method for unsteady laminar and turbulent flows, (2018 AIAA Aerospace Sciences Meeting. 2018 AIAA Aerospace Sciences Meeting, January, Kissimmee, Florida (2018)) |
| [25] | Dumbser, M.; Zanotti, O.; Loubère, R.; Diot, S., A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws, J. Comput. Phys., 278, 47-75 (2014) · Zbl 1349.65448 |
| [26] | Dumbser, M.; Loubère, R., A simple robust and accurate a posteriori sub-cell finite volume limiter for the discontinuous Galerkin method on unstructured meshes, J. Comput. Phys., 319, 163-199 (2016) · Zbl 1349.65447 |
| [27] | Titarev, V.; Toro, E., ADER: arbitrary high order Godunov approach, J. Sci. Comput., 17, 609-618 (2002) · Zbl 1024.76028 |
| [28] | Schwartzkopff, T.; Munz, C.; Toro, E., ADER: a high order approach for linear hyperbolic systems in 2D, J. Sci. Comput., 17, 231-240 (2002) · Zbl 1022.76034 |
| [29] | Dumbser, M.; Balsara, D. S.; Toro, E. F.; Munz, C. D., A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes, J. Comput. Phys., 227, 8209-8253 (2008) · Zbl 1147.65075 |
| [30] | Dumbser, M., Arbitrary high order \(\operatorname{P}_{\operatorname{N}} \operatorname{P}_{\operatorname{M}}\) schemes on unstructured meshes for the compressible Navier-Stokes equations, Comput. Fluids, 39, 60-76 (2010) · Zbl 1242.76161 |
| [31] | Xu, K., A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comput. Phys., 171, 289-335 (2001) · Zbl 1058.76056 |
| [32] | Li, Q. B.; Fu, S.; Xu, K., A compressible Navier-Stokes flow solver with scalar transport, J. Comput. Phys., 204, 692-714 (2005) · Zbl 1329.76310 |
| [33] | Li, Q. B.; Fu, S., On the multidimensional gas-kinetic BGK scheme, J. Comput. Phys., 220, 532-548 (2006) · Zbl 1108.76045 |
| [34] | Xu, K.; Mao, M. L.; Tang, L., A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow, J. Comput. Phys., 203, 405-421 (2005) · Zbl 1143.76553 |
| [35] | Li, Q. B.; Fu, S., A high-order accurate gas-kinetic BGK scheme, (5th International Conference on Computational Fluid Dynamics. 5th International Conference on Computational Fluid Dynamics, July 7-11, Seoul, Korea (2008)) |
| [36] | Li, Q. B.; Xu, K.; Fu, S., A high-order gas-kinetic Navier-Stokes flow solver, J. Comput. Phys., 229, 6715-6731 (2010) · Zbl 1425.76166 |
| [37] | Li, Q. B.; Xu, K.; Fu, S., On the high-order multidimensional gas-kinetic scheme, (7th International Conference on Computational Fluid Dynamics. 7th International Conference on Computational Fluid Dynamics, July 9-13, Big Island, Hawaii (2012)) |
| [38] | Pan, L.; Xu, K., A compact third-order gas-kinetic scheme for compressible Euler and Navier-Stokes equations, Commun. Comput. Phys., 18, 985-1011 (2015) · Zbl 1373.76184 |
| [39] | Pan, L.; Xu, K., A third-order compact gas-kinetic scheme on unstructured meshes for compressible Navier-Stokes solutions, J. Comput. Phys., 318, 327-348 (2016) · Zbl 1349.76517 |
| [40] | Liu, N.; Xu, X.; Chen, Y., High-order spectral volume scheme for multi-component flows using non-oscillatory kinetic flux, Comput. Fluids, 152, 120-133 (2017) · Zbl 1390.76476 |
| [41] | Ren, X.; Xu, K.; Wei, S.; Gu, C., A multi-dimensional high-order discontinuous Galerkin method based on gas kinetic theory for viscous, J. Comput. Phys., 292, 176-193 (2015) · Zbl 1349.65479 |
| [42] | Li, J.; Zhong, C.; Zhuo, C., A third order gas-kinetic scheme for unstructured grid, Comput. Math. Appl., 78, 92-109 (2019) · Zbl 1442.76105 |
| [43] | Zhang, C.; Li, Q. B., A third-order subcell finite volume gas-kinetic scheme for the Euler and Navier-Stokes equations on triangular meshes, J. Comput. Phys., 436, Article 110245 pp. (2021) · Zbl 07513839 |
| [44] | Zhang, C.; Li, Q. B.; Fu, S.; Wang, Z. J., A third-order gas-kinetic CPR method for the Euler and Navier-Stokes equations on triangular meshes, J. Comput. Phys., 363, 329-353 (2018) · Zbl 1392.76057 |
| [45] | Liu, N.; Tang, H., A high-order accurate gas-kinetic scheme for one and two-dimensional flow simulation, Commun. Comput. Phys., 15, 911-943 (2014) · Zbl 1373.76259 |
| [46] | Li, J.; Du, Z., A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers, I: hyperbolic conservation laws, SIAM J. Sci. Comput., 38, 5, A3046-A3069 (2016) · Zbl 1395.65040 |
| [47] | Du, Z.; Li, J., A two-stage fourth order time-accurate discretization for Lax-Wendroff type flow solvers II: high order numerical boundary conditions, J. Comput. Phys., 369, 125-147 (2018) · Zbl 1395.65037 |
| [48] | Li, J., Two-stage fourth order: temporal-spatial coupling in computational fluid dynamics (CFD), Adv. Aerodyn., 1-3 (2019) |
| [49] | Cheng, J.; Du, Z.; Lei, X.; Wang, Y.; Li, J., A two-stage fourth-order discontinuous Galerkin method based on the GRP solver for the compressible Euler equations, Comput. Fluids, 181, 248-258 (2019) · Zbl 1410.76161 |
| [50] | Pan, L.; Xu, K.; Li, Q. B.; Li, J., An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations, J. Comput. Phys., 326, 197-221 (2016) · Zbl 1373.76032 |
| [51] | Zhao, F.; Ji, X.; Shyy, W.; Xu, K., Compact higher-order gas-kinetic schemes with spectral-like resolution for compressible flow simulations, Adv. Aerodyn., 1-13 (2019) |
| [52] | Ji, X.; Zhao, F.; Shyy, W.; Xu, K., A family of high-order gas-kinetic schemes and its comparison with Riemann solver based high-order methods, J. Comput. Phys., 356, 150-173 (2018) · Zbl 1380.76061 |
| [53] | Seal, D. C.; Güclü, Y.; Christlieb, A. J., High-order multiderivative time integrators for hyperbolic conservation laws, J. Sci. Comput., 60, 1, 101-140 (2014) · Zbl 1304.65223 |
| [54] | Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision processes in gases I: small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94, 511-525 (1954) · Zbl 0055.23609 |
| [55] | Xu, K., Direct Modeling for Computational Fluid Dynamics: Construction and Application of Unified Gas-Kinetic Schemes (2015), World Scientific · Zbl 1309.76003 |
| [56] | Fu, G.; Shu, C.-W., A new troubled-cell indicator for discontinuous Galerkin methods for hyperbolic conservation laws, J. Comput. Phys., 347, 305-327 (2017) · Zbl 1380.65262 |
| [57] | Barth, T. J.; Jespersen, D. C., The design and application of upwind schemes on unstructured meshes, 1-12 (1989), AIAA Paper 89-0366 |
| [58] | Carpenter, M. H.; Kennedy, C., Fourth-order 2N-storage Runge-Kutta schemes (1994), NASA TM-109112 |
| [59] | Zhang, C.; Li, Q. B., A fourth-order gas-kinetic CPR method for the Navier-Stokes equations on unstructured meshes, (10th International Conference on Computational Fluid Dynamics. 10th International Conference on Computational Fluid Dynamics, July 9-13, Barcelona, Spain (2018)) |
| [60] | Shu, C.-W.; Osher, S., Efficient implementation of essentially nonoscillatory shock-capturing schemes II, J. Comput. Phys., 83, 32-78 (1989) · Zbl 0674.65061 |
| [61] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 115-173 (1984) · Zbl 0573.76057 |
| [62] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2013), Springer Science & Business Media |
| [63] | Boscheri, W.; Dumbser, M., An efficient quadrature-free formulation for high-order arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes on unstructured meshes, J. Sci. Comput., 66, 240-274 (2016) · Zbl 1338.65219 |
| [64] | Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys., 228, 2391-2425 (2009) · Zbl 1156.76434 |
| [65] | Daru, V.; Tenaud, C., High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations, J. Comput. Phys., 193, 563-594 (2004) · Zbl 1109.76338 |
| [66] | Zhou, G.; Xu, K.; Liu, F., Grid-converged solution and analysis of the unsteady viscous flow in a two-dimensional shock tube, Phys. Fluids, 30, Article 016102 pp. (2018) |
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.
