Numerical investigation of a viscous regularization of the Euler equations by entropy viscosity. (English) Zbl 1439.76025
Summary: The Navier-Stokes viscous fluxes are a well-known viscous regularization of the Euler equations. However, since these fluxes do not add any viscosity to the mass equation, the positivity of density is violated. This paper investigates a new class of viscous regularization of the Euler equations, which was recently proposed by J.-L. Guermond and B. Popov [SIAM J. Appl. Math. 74, No. 2, 284–305 (2014; Zbl 1446.76147)]. In contrast to the Navier-Stokes fluxes, the new regularization adds a viscous term to the mass equation. Since non-physical viscous terms are used, it is important to show that the exact solution’s properties, such as the location of shocks, contact and rarefaction waves are not violated. The present study concerns a careful numerical investigation of the new viscous regularization in a number of well-known 1D and 2D benchmark problems. Also, a direct numerical comparison with respect to the physical Navier-Stokes regularization is shown. The numerical tests show that the entropy viscosity method can achieve high order accuracy for any polynomial degrees. Detailed algorithms for the implementation of a slip wall boundary condition are presented in a weak and a strong form.
Keywords:
entropy viscosity; conservation laws; compressible flow; Euler equations; finite elements; nonlinear stabilizationCitations:
Zbl 1446.76147References:
| [1] | Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics. IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg., 58, 3, 329-336 (1986) · Zbl 0587.76120 |
| [2] | Johnson, C.; Szepessy, A.; Hansbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. Comp., 54, 189, 107-129 (1990) · Zbl 0685.65086 |
| [3] | Nazarov, M., Convergence of a residual based artificial viscosity finite element method, Comput. Math. Appl., 65, 4, 616-626 (2013) · Zbl 1319.65098 |
| [4] | Nazarov, M.; Hoffman, J., Residual-based artificial viscosity for simulation of turbulent compressible flow using adaptive finite element methods, Internat. J. Numer. Methods Fluids, 71, 3, 339-357 (2013) · Zbl 1430.76314 |
| [5] | Marras, S.; Nazarov, M.; Giraldo, F. X., Stabilized high-order Galerkin methods based on a parameter-free dynamic SGS model for LES, J. Comput. Phys., 301, 77-101 (2015) · Zbl 1349.76127 |
| [6] | Barter, G. E.; Darmofal, D. L., Shock capturing with PDE-based artificial viscosity for DGFEM. I. Formulation, J. Comput. Phys., 229, 5, 1810-1827 (2010) · Zbl 1329.76153 |
| [7] | Reisner, J.; Serencsa, J.; Shkoller, S., A space-time smooth artificial viscosity method for nonlinear conservation laws, J. Comput. Phys., 235, 912-933 (2013) |
| [8] | Guermond, J.-L.; Pasquetti, R., Entropy-based nonlinear viscosity for Fourier approximations of conservation laws, C. R. Math. Acad. Sci. Paris, 346, 13-14, 801-806 (2008) · Zbl 1145.65079 |
| [9] | Guermond, J.-L.; Pasquetti, R.; Popov, B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 230, 11, 4248-4267 (2011) · Zbl 1220.65134 |
| [10] | P. Persson, J. Peraire, Sub-Cell Shock Capturing for Discontinuous Galerkin Methods, AIAA Aerospace Sciences Meeting and Exhibit, 44th.; P. Persson, J. Peraire, Sub-Cell Shock Capturing for Discontinuous Galerkin Methods, AIAA Aerospace Sciences Meeting and Exhibit, 44th. |
| [11] | Zingan, V.; Guermond, J.-L.; Morel, J.; Popov, B., Implementation of the entropy viscosity method with the discontinuous Galerkin method, Comput. Methods Appl. Mech. Engrg., 253, 479-490 (2013) · Zbl 1297.76109 |
| [12] | Guermond, J.-L.; Pasquetti, R.; Popov, B., From suitable weak solutions to entropy viscosity, J. Sci. Comput., 49, 1, 35-50 (2011) · Zbl 1432.76080 |
| [13] | Nazarov, M., Adaptive Algorithms and High Order Stabilization for Finite Element Computation of Turbulent Compressible Flow (2011), KTH, Numerical Analysis, NA, QC 20110627 |
| [14] | Guermond, J. L.; Pasquetti, R., Entropy viscosity method for high-order approximations of conservation laws, (Spectral and High Order Methods for Partial Differential Equations. Spectral and High Order Methods for Partial Differential Equations, Lect. Notes Comput. Sci. Eng., vol. 76 (2011), Springer: Springer Heidelberg), 411-418 · Zbl 1216.65136 |
| [15] | Guermond, J.-L.; Popov, B., Viscous regularization of the Euler equations and entropy principles, SIAM J. Appl. Math., 74, 2, 284-305 (2014) · Zbl 1446.76147 |
| [16] | Delchini, M. O.; Ragusa, J. C.; Berry, R. A., Entropy-based viscous regularization for the multi-dimensional Euler equations in low-Mach and transonic flows, Comput. & Fluids, 118, 225-244 (2015) · Zbl 1390.76255 |
| [17] | Layton, W., Weak imposition of “no-slip” conditions in finite element methods, Comput. Math. Appl., 38, 5-6, 129-142 (1999) · Zbl 0953.76050 |
| [18] | John, V., Slip with friction and penetration with resistance boundary conditions for the Navier-Stokes equations—numerical tests and aspects of the implementation, J. Comput. Appl. Math., 147, 2, 287-300 (2002) · Zbl 1021.76028 |
| [19] | Harten, A.; Lax, P. D.; Levermore, C. D.; Morokoff, W. J., Convex entropies and hyperbolicity for general Euler equations, SIAM J. Numer. Anal., 35, 6, 2117-2127 (1998), (electronic). http://dx.doi.org/10.1137/S0036142997316700 · Zbl 0922.35089 |
| [20] | Serre, D., (Systèmes de lois de conservation. I. Systèmes de lois de conservation. I, Fondations. [Foundations] (1996), Diderot Editeur: Diderot Editeur Paris), Hyperbolicité, entropies, ondes de choc. [Hyperbolicity, entropies, shock waves] · Zbl 0930.35002 |
| [21] | Harten, A., On the symmetric form of systems of conservation laws with entropy, J. Comput. Phys., 49, 1, 151-164 (1983) · Zbl 0503.76088 |
| [22] | Guermond, J.-L.; Nazarov, M.; Popov, B., Implementation of the Entropy Viscosity Method, Tech. Rep. 4015 (2011), KTH, Numerical Analysis, NA, qC 20110720 |
| [23] | Nazarov, M.; Hoffman, J., On the stability of the dual problem for high Reynolds number flow past a circular cylinder in two dimensions, SIAM J. Sci. Comput., 34, 4, A1905-A1924 (2012) · Zbl 1250.76132 |
| [24] | Logg, A.; Mardal, K.-A.; Wells, G. N., Automated Solution of Differential Equations by the Finite Element Method (2012), Springer · Zbl 1247.65105 |
| [25] | Guermond, J.-L.; Nazarov, M., A maximum-principle preserving \(C^0\) finite element method for scalar conservation equations, Comput. Methods Appl. Mech. Engrg., 272, 198-213 (2014) · Zbl 1296.65133 |
| [26] | Süli, E.; Mayers, D. F., An Introduction to Numerical Analysis (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1033.65001 |
| [27] | Gottlieb, S.; Shu, C.-W.; Tadmor, E., Strong stability-preserving high-order time discretization methods, SIAM Rev., 43, 1, 89-112 (electronic) (2001) · Zbl 0967.65098 |
| [28] | Ern, A.; Guermond, J.-L., Weighting the edge stabilization, SIAM J. Numer. Anal., 51, 3, 1655-1677 (2013) · Zbl 1280.65095 |
| [29] | Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (2009), Springer-Verlag: Springer-Verlag Berlin, a practical introduction. http://dx.doi.org/10.1007/b79761 · Zbl 1227.76006 |
| [30] | Guermond, J. L.; Popov, B., Invariant domains and first-order continuous finite element approximation for hyperbolic systems, SIAM J. Numer. Anal., 54, 4, 2466-2489 (2016), arXiv:http://dx.doi.org/10.1137/16M1074291 · Zbl 1346.65050 |
| [31] | Fletcher, C. A.J., (Computational Techniques for Fluid Dynamics. 1. Computational Techniques for Fluid Dynamics. 1, Springer Series in Computational Physics (1988), Springer-Verlag: Springer-Verlag Berlin), Fundamental and general techniques · Zbl 0706.76001 |
| [32] | Wesseling, P., (Principles of Computational Fluid Dynamics. Principles of Computational Fluid Dynamics, Springer Series in Computational Mathematics, vol. 29 (2001), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0960.76002 |
| [33] | Kuzmin, D.; Möller, M., Algebraic flux correction. II. Compressible Euler equations, (Flux-Corrected Transport. Flux-Corrected Transport, Sci. Comput. (2005), Springer: Springer Berlin), 207-250 · Zbl 1328.76042 |
| [34] | Hoffman, J.; Jansson, J.; Degirmenci, C.; Jansson, N.; Nazarov, M., Unicorn: a unified continuum mechanics solver, (Logg, A.; Mardal, K.-A.; Wells, G. N., Automated Solution of Differential Equations by the Finite Element Method (2011), Springer) |
| [35] | Hoffman, J.; Jansson, J.; Vilela de Abreu, R.; Degirmenci, N. C.; Jansson, N.; Müller, K.; Nazarov, M.; Spühler, J. H., Unicorn: parallel adaptive finite element simulation of turbulent flow and fluid-strucure interaction for deforming domains and complex geometry, Comput. & Fluids, 80, 310-319 (2013) · Zbl 1284.76223 |
| [36] | DeVore, R. A., Nonlinear approximation, (Acta Numerica, 1998. Acta Numerica, 1998, Acta Numer., vol. 7 (1998), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 51-150 · Zbl 0931.65007 |
| [37] | Liska, R.; Wendroff, B., Comparison of several difference schemes on 1D and 2D test problems for the Euler equations, SIAM J. Sci. Comput., 25, 3, 995-1017 (2003), (electronic). http://dx.doi.org/10.1137/S1064827502402120 · Zbl 1096.65089 |
| [38] | Emery, A. F., An evaluation of several differencing methods for inviscid fluid flow problems, J. Comput. Phys., 2, 306-331 (1968) · Zbl 0155.21102 |
| [39] | Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 1, 115-173 (1984) · Zbl 0573.76057 |
| [40] | Banks, J. W.; Henshaw, W. D.; Schwendeman, D. W.; Kapila, A. K., A study of detonation propagation and diffraction with compliant confinement, Combust. Theory Model., 12, 4, 769-808 (2008) · Zbl 1144.80379 |
| [41] | Niederhaus, J.; Ranjan, D.; Oakley, J.; Anderson, M.; Greenough, J.; Bonazza, R., Computations in 3D for shock-induced distortion of a light spherical gas inhomogeneity, (Hannemann, K.; Seiler, F., Shock Waves (2009), Springer: Springer Berlin, Heidelberg), 1169-1174 |
| [42] | Langseth, J. O.; LeVeque, R. J., A wave propagation method for three-dimensional hyperbolic conservation laws, J. Comput. Phys., 165, 1, 126-166 (2000) · Zbl 0967.65095 |
| [43] | Schulz-Rinne, C. W., Classification of the Riemann problem for two-dimensional gas dynamics, SIAM J. Math. Anal., 24, 1, 76-88 (1993) · Zbl 0811.35082 |
| [44] | Schroll, H. J.; Svensson, F., A bi-hyperbolic finite volume method on quadrilateral meshes, J. Sci. Comput., 26, 2, 237-260 (2006) · Zbl 1203.76096 |
| [45] | Holden, H.; Lie, K.-A.; Risebro, N. H., An unconditionally stable method for the Euler equations, J. Comput. Phys., 150, 1, 76-96 (1999) · Zbl 0922.76251 |
| [46] | Boris, J. P.; Book, D. L., Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works [J. Comput. Phys. 11 (1) (1973) 38-69], J. Comput. Phys., 135, 2, 170-186 (1997), With an introduction by Steven T. Zalesak, Commemoration of the 30th anniversary of J. Comput. Phys |
| [47] | Zalesak, S. T., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 3, 335-362 (1979) · Zbl 0416.76002 |
| [48] | Guermond, J.-L.; Nazarov, M.; Popov, B.; Yang, Y., A second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations, SIAM J. Numer. Anal., 52, 4, 2163-2182 (2014) · Zbl 1302.65225 |
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.
