Mass, momentum and energy identical-relation-preserving scheme for the Navier-Stokes equations with variable density. (English) Zbl 1538.76060
Summary: In this paper, we study an unconditionally stable and decoupled identical-relation-preserving scheme for solving the incompressible Navier-Stokes equations with variable density. By introducing power- and exponential-type scalar auxiliary variables to define the system’s energy and to balance the incompressible condition’s influence respectively, we first convert the Navier-Stokes equations with variable density into an equivalent form. Then, with the help of the equations of scalar auxiliary variables, we construct a linear decoupled fully discrete finite element scheme, which can be implemented efficiently. Moreover, the mass, momentum and energy identical relations which are analogous to that for the continuous equations are proved uniformly regardless of the discretization parameters. Both lower- and upper-bound of the numerical density measured in \(L^2\)-norm are deduced, too. Finally, we report some numerical examples to verify the correctness and efficiency of the proposed scheme, including an extension of a coupled nonlinear model with temperature.
MSC:
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 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 |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Navier-Stokes equations with variable density; identical-relation-preserving scheme; scalar auxiliary variable; finite element method; unconditional stabilitySoftware:
FreeFem++References:
| [1] | Almgren, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M. L., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, J. Comput. Phys., 142, 1-46 (1998) · Zbl 0933.76055 |
| [2] | An, R., Error analysis of a new fractional-step method for the incompressible Navier-Stokes equations with variable density, J. Sci. Comput., 84, Article 3 pp. (2020) · Zbl 1450.65115 |
| [3] | An, R., Error analysis of a time-splitting method for incompressible flows with variable density, Appl. Numer. Math., 150, 384-395 (2020) · Zbl 1448.76056 |
| [4] | Antontsev, S. N.; Kazhikhov, A. V.; Monakhov, V. N., Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, Studies in Mathematics and Its Applications, vol. 22 (1990), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0696.76001 |
| [5] | Bruus, H., Theoretical Microfluidics (2008), Oxford University Press |
| [6] | Cai, W.; Li, B.; Li, Y., Error analysis of a fully discrete finite element method for variable density incompressible flows in two dimensions, ESAIM: Math. Model. Numer. Anal., 55, S103-S147 (2021) · Zbl 1477.65156 |
| [7] | Calgaro, C.; Creuse, E.; Goudon, T., An hybrid finite volume-finite element method for variable density incompressible flows, J. Comput. Phys., 227, 4671-4696 (2008) · Zbl 1137.76037 |
| [8] | Charnyi, S.; Heister, T.; Olshanskii, M. A.; Rebholz, L. G., On conservation laws of Navier-Stokes Galerkin discretizations, J. Comput. Phys., 337, 289-308 (2017) · Zbl 1415.65222 |
| [9] | Chen, H.; Mao, J.; Shen, J., Error estimate of Gauge-Uzawa methods for incompressible flows with variable density, J. Comput. Appl. Math., 364, Article 112321 pp. (2020) · Zbl 1427.76124 |
| [10] | Chen, H.; Mao, J.; Shen, J., Optimal error estimates for the scalar auxiliary variable finite-element schemes for gradient flows, Numer. Math., 145, 167-196 (2020) · Zbl 1440.65192 |
| [11] | Danchin, R., Density-dependent incompressible fluids in bounded domains, J. Math. Fluid Mech., 8, 333-381 (2006) · Zbl 1142.76354 |
| [12] | Desmons, F.; Coquerelle, M., A generalized high-order momentum preserving (HOMP) method in the one-fluid model for incompressible two phase flows with high density ratio, J. Comput. Phys., 437, Article 110322 pp. (2021) · Zbl 07505907 |
| [13] | Fujita, H.; Kato, T., On the Navier-Stokes initial value problem, I, Arch. Ration. Mech. Anal., 16, 269-315 (1964) · Zbl 0126.42301 |
| [14] | Guermond, J. L.; Minev, P.; Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195, 44-47, 6011-6045 (2006) · Zbl 1122.76072 |
| [15] | Guermond, J. L.; Quartapelle, L., A projection FEM for variable density incompressible flows, J. Comput. Phys., 165, 167-188 (2000) · Zbl 0994.76051 |
| [16] | Guermond, J. L.; Salgado, A., A splitting method for incompressible flows with variable density based on a pressure Poisson equation, J. Comput. Phys., 228, 2834-2846 (2009) · Zbl 1159.76028 |
| [17] | Guermond, J. L.; Salgado, A. J., Error analysis of a fractional time-stepping technique for incompressible flows with variable density, SIAM J. Numer. Anal., 49, 917-944 (2011) · Zbl 1241.76318 |
| [18] | He, Y., Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations, Math. Comput., 251, 1201-1216 (2005) · Zbl 1065.35025 |
| [19] | Hecht, F., New development in freefem++, J. Numer. Math., 20, 251-265 (2012) · Zbl 1266.68090 |
| [20] | Horváth, T. L.; Rhebergen, S., An exactly mass conserving space-time embedded-hybridized discontinuous Galerkin method for the Navier-Stokes equations on moving domains, J. Comput. Phys., 417, Article 109577 pp. (2020) · Zbl 1437.76021 |
| [21] | Ingimarson, S., An energy, momentum, and angular momentum conserving scheme for a regularization model of incompressible flow, J. Numer. Math., 30, 1-22 (2022) · Zbl 1537.65133 |
| [22] | Ladyzhenskaya, O.; Solonnikov, V., Unique solvability of an initial- and boundary-value problem for viscous incompressible inhomogeneous fluids, J. Sov. Math., 9, 697-749 (1978) · Zbl 0401.76037 |
| [23] | Li, B.; Qiu, W.; Yang, Z., A convergent post-processed discontinuous Galerkin method for incompressible flow with variable density, J. Sci. Comput., 91, Article 2 pp. (2022) · Zbl 1484.65225 |
| [24] | Li, M.; Cheng, Y.; Shen, J.; Zhang, X., A bound-preserving high order scheme for variable density incompressible Navier-Stokes equations, J. Comput. Phys., 425, Article 109906 pp. (2021) · Zbl 07508502 |
| [25] | Li, X.; Shen, J., Stability and error estimates of the SAV Fourier-spectral method for the phase field crystal equation, Adv. Comput. Math., 46, Article 48 pp. (2020) · Zbl 1442.65294 |
| [26] | Li, X.; Shen, J., On a SAV-MAC scheme for the Cahn-Hilliard-Navier-Stokes phase-field model and its error analysis for the corresponding Cahn-Hilliard-Stokes case, Math. Models Methods Appl. Sci., 30, 2263-2297 (2020) · Zbl 1471.65106 |
| [27] | Li, X.; Shen, J.; Liu, Z., New SAV-pressure correction methods for the Navier-Stokes equations: stability and error analysis, Math. Comput., 141-167 (2022) · Zbl 1479.35625 |
| [28] | Li, Y.; An, R., Unconditionally optimal error analysis of a linear Euler FEM scheme for the Navier-Stokes equations with mass diffusion, J. Sci. Comput., 90, Article 47 pp. (2022) · Zbl 1481.35337 |
| [29] | Li, Y.; Mei, L.; Ge, J.; Shi, F., A new fractional time-stepping method for variable density incompressible flows, J. Comput. Phys., 242, 124-137 (2013) · Zbl 1311.76068 |
| [30] | Lin, L.; Yang, Z.; Dong, S., Numerical approximation of incompressible Navier-Stokes equations based on an auxiliary energy variable, J. Comput. Phys., 388, 1-22 (2019) · Zbl 1452.76093 |
| [31] | Lions, P. L., Mathematical Topics in Fluid Mechanics, Volume 1: Incompressible Models (1996), Oxford Press: Oxford Press Oxford · Zbl 0866.76002 |
| [32] | Liu, C.; Walkington, N. J., Convergence of numerical approximations of the incompressible Navier-Stokes equations with variable density and viscosity, SIAM J. Numer. Anal., 45, 1287-1304 (2007) · Zbl 1138.76048 |
| [33] | Manzanero, J.; Rubio, G.; Kopriva, D. A.; Ferrer, E.; Valero, E., An entropy-stable discontinuous Galerkin approximation for the incompressible Navier-Stokes equations with variable density and artificial compressibility, J. Comput. Phys., 408, Article 109241 pp. (2020) · Zbl 07505602 |
| [34] | Olshanskii, M. A.; Rebholz, L. G., Longer time accuracy for incompressible Navier-Stokes simulations with the EMAC formulation, Comput. Methods Appl. Mech. Eng., 372, Article 113369 pp. (2020) · Zbl 1506.76021 |
| [35] | Ortega-Torres, E.; Braz e. Silva, P.; Rojas-Medar, M., Analysis of an iterative method for variable density incompressible fluids, Ann. Univ. Ferrara, 55, Article 129 pp. (2009) · Zbl 1185.35171 |
| [36] | Palha, A.; Gerritsma, M., A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations, J. Comput. Phys., 328, 200-220 (2017) · Zbl 1406.76064 |
| [37] | Prohl, A., Projection and Qusi-Compressibility Methods for Solving the Incompressible Navier-Stokes Equations, Advances in Numerical Mathematics (1997), Springer Fachmedien Wiesbaden · Zbl 0874.76002 |
| [38] | Puckett, E. G.; Almgren, A. S.; Bell, J. B.; Marcus, D. L.; Rider, W. J., A high-order projection method for tracking fluid interfaces in variable density incompressible flows, J. Comput. Phys., 130, 269-282 (1997) · Zbl 0872.76065 |
| [39] | Pyo, J. H.; Shen, J., Gauge-Uzawa methods for incompressible flows with variable density, J. Comput. Phys., 221, 181-197 (2007) · Zbl 1109.76037 |
| [40] | Shen, J.; Zheng, N., Efficient and unconditional energy stable schemes for the micropolar Navier-Stokes equations, CSIAM Trans. Appl. Math., 3, 57-81 (2022) |
| [41] | Shen, J.; Xu, J.; Yang, J., A new class of efficient and robust energy stable schemes for gradient flows, SIAM Rev., 61, 474-506 (2019) · Zbl 1422.65080 |
| [42] | Szewc, K.; Pozorski, J.; Taniere, A., Modeling of natural convection with smoothed particle hydrodynamics: non-Boussinesq formulation, Int. J. Heat Mass Transf., 54, 23, 4807-4816 (2011) · Zbl 1226.80024 |
| [43] | Wu, J.; Huang, P.; Feng, X., The characteristic variational multiscale method for time dependent conduction-convection problems, Int. Commun. Heat Transf., Part A, Appl., 68, 777-796 (2015) |
| [44] | Wu, J.; Shen, J.; Feng, X., Unconditionally stable Gauge-Uzawa finite element schemes for incompressible natural convection problems with variable density, J. Comput. Phys., 348, 776-789 (2017) · Zbl 1380.76132 |
| [45] | Wu, J.; Wei, L.; Feng, X., Novel fractional time-stepping algorithms for natural convection problems with variable density, Appl. Numer. Math., 151, 64-84 (2020) · Zbl 1448.76145 |
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.
