×

zbMATH — the first resource for mathematics

Entropy-stable, high-order summation-by-parts discretizations without interface penalties. (English) Zbl 1434.65184
The author combines nodal summation-by-parts operators as in continuous Galerkin methods to construct entropy conservative semidiscretizations of hyperbolic conservation laws. A local projection stabilization is used to reduce possible oscillations and render the scheme entropy dissipative. Numerical results for the linear advection and compressible Euler equations are presented.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods 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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baiocchi, C.; Brezzi, F.; Franca, Lp, Virtual bubbles and Galerkin-least-squares type methods (Ga.L.S.), Comput. Methods Appl. Mech. Eng., 105, 1, 125-141 (1993) · Zbl 0772.76033
[2] Barth, T.J.: Numerical methods for gasdynamic systems on unstructured meshes. In: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, pp. 195-285. Springer (1999) · Zbl 0969.76040
[3] Becker, R.; Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38, 4, 173-199 (2001) · Zbl 1008.76036
[4] Bezanson, J.; Edelman, A.; Karpinski, S.; Shah, Vb, Julia: a fresh approach to numerical computing, SIAM Rev., 59, 1, 65-98 (2017) · Zbl 1356.68030
[5] Braack, M.; Lube, G., Finite elements with local projection stabilization for incompressible flow problems, J. Comput. Math., 27, 2-3, 116-147 (2009) · Zbl 1212.65407
[6] Brooks, An; Hughes, Tjr, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 1-3, 199-259 (1982) · Zbl 0497.76041
[7] Burman, E.; Fernández, Ma; Hansbo, P., Continuous interior penalty finite element method for Oseen’s equations, SIAM J. Numer. Anal., 44, 3, 1248-1274 (2006) · Zbl 1344.76049
[8] Burman, E.; Hansbo, P., Edge stabilization for galerkin approximations of convection-diffusion-reaction problems, Comput. Methods Appl. Mech. Eng., 193, 15-16, 1437-1453 (2004) · Zbl 1085.76033
[9] Carpenter, Mh; Fisher, Tc; Nielsen, Ej; Frankel, Sh, Entropy stable spectral collocation schemes for the Navier-Stokes equations: discontinuous interfaces, SIAM J. Sci. Comput., 36, 5, B835-B867 (2014) · Zbl 1457.65140
[10] Chandrashekar, P., Kinetic energy preserving and entropy stable finite volume schemes for compressible euler and Navier-Stokes equations, Commun. Comput. Phys., 14, 5, 1252-1286 (2015) · Zbl 1373.76121
[11] Chen, T.; Shu, Cw, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws, J. Comput. Phys., 345, 427-461 (2017) · Zbl 1380.65253
[12] Cockburn, B.; Hou, S.; Shu, Cw, 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
[13] Cools, R., Monomial cubature rules since “stroud”: a compilation—part 2, J. Comput. Appl. Math., 112, 12, 21-27 (1999) · Zbl 0954.65021
[14] Craig Penner, D., Zingg, D.W.: High-order artificial dissipation operators possessing the Summation-By-parts property. In: 2018 Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics (2018). 10.2514/6.2018-4165
[15] Crean, J.; Hicken, Je; Del Rey Fernández, Dc; Zingg, Dw; Carpenter, Mh, Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements, J. Comput. Phys., 356, 410-438 (2018) · Zbl 1380.76080
[16] Crean, J., Panda, K., Ashley, A., Hicken, J.E.: Investigation of stabilization methods for multi-dimensional summation-by-parts discretizations of the Euler equations. In: 54th AIAA Aerospace Sciences Meeting, p. 13. San Diego, California, United States. AIAA 2016-1328 (2016). 10.2514/6.2016-1328
[17] Dafermos, Cm, Hyperbolic Conservation Laws in Continuum Physics (2010), Berlin: Springer, Berlin · Zbl 1196.35001
[18] Del Rey Fernández, Dc; Boom, Pd; Zingg, Dw, A generalized framework for nodal first derivative summation-by-parts operators, J. Comput. Phys., 266, 1, 214-239 (2014) · Zbl 1311.65002
[19] Del Rey Fernández, Dc; Crean, J.; Carpenter, Mh; Hicken, Je, Staggered-grid entropy-stable multidimensional summation-by-parts discretizations on curvilinear coordinates, J. Comput. Phys., 392, 161-186 (2019)
[20] Del Rey Fernández, Dc; Hicken, Je; Zingg, Dw, Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. Fluids, 95, 22, 171-196 (2014) · Zbl 1390.65064
[21] Del Rey Fernández, Dc; Hicken, Je; Zingg, Dw, Simultaneous approximation terms for multi-dimensional summation-by-parts operators, J. Sci. Comput., 74, 83-110 (2017) · Zbl 06865000
[22] Douglas, J.; Dupont, T.; Glowinski, R.; Lions, Jl, Interior penalty procedures for elliptic and parabolic Galerkin methods computing methods in applied sciences, Computing Methods in Applied Sciences. Lecture Notes in Physics, chap. 6, 207-216 (1976), Berlin: Springer, Berlin
[23] Fisher, T.C.: High-order l2 stable multi-domain finite difference method for compressible flows. Ph.D. thesis, Purdue University (2012)
[24] Fisher, Tc; Carpenter, Mh, High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains, J. Comput. Phys., 252, 518-557 (2013) · Zbl 1349.65293
[25] Fisher, Tc; Carpenter, Mh; Nordström, J.; Yamaleev, Nk; Swanson, C., Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: theory and boundary conditions, J. Comput. Phys., 234, 353-375 (2013) · Zbl 1284.65102
[26] Gassner, Gj, A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., 35, 3, A1233-A1253 (2013) · Zbl 1275.65065
[27] Hartmann, R., Adjoint consistency analysis of discontinuous Galerkin discretizations, SIAM J. Numer. Anal., 45, 6, 2671-2696 (2007) · Zbl 1189.76341
[28] Hesthaven, Js; Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (2008), New York: Springer, New York · Zbl 1134.65068
[29] Hicken, Je; Del Rey Fernández, Dc; Zingg, Dw, Multi-dimensional summation-by-parts operators: general theory and application to simplex elements, SIAM J. Sci. Comput., 38, 4, A1935-A1958 (2016) · Zbl 1382.65355
[30] Hicken, Je; Zingg, Dw, Summation-by-parts operators and high-order quadrature, J. Comput. Appl. Math., 237, 1, 111-125 (2013) · Zbl 1263.65025
[31] Hicken, Je; Zingg, Dw, Dual consistency and functional accuracy: a finite-difference perspective, J. Comput. Phys., 256, 161-182 (2014) · Zbl 1349.65559
[32] Hughes, Tjr, Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Eng., 127, 387-401 (1995) · Zbl 0866.76044
[33] Hughes, Tjr; Franca, Lp; Hulbert, Gm, A new finite element formulation for computational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Eng., 73, 2, 173-189 (1989) · Zbl 0697.76100
[34] Hughes, Tjr; Franca, Lp; Mallet, M., A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Navier-Stokes equations and the second law of thermodymaics, Comput. Methods Appl. Mech. Eng., 54, 2, 223-234 (1986) · Zbl 0572.76068
[35] Hunter, Jd, Matplotlib: a 2D graphics environment, Comput. Sci. Eng., 9, 3, 90-95 (2007)
[36] Ismail, F.; Roe, Pl, Affordable, entropy-consistent euler flux functions II: entropy production at shocks, J. Comput. Phys., 228, 15, 5410-5436 (2009) · Zbl 1280.76015
[37] Jameson, A., Schmidt, W., Turkel, E.: Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes. In: 14th Fluid and Plasma Dynamics Conference. Palo Alto, CA (1981)
[38] Jones, E., Oliphant, T., Peterson, P., et al.: SciPy: open source scientific tools for Python (2001-). http://www.scipy.org/
[39] Liu, Y.; Vinokur, M., Exact integrations of polynomials and symmetric quadrature formulas over arbitrary polyhedral grids, J. Comput. Phys., 140, 1, 122-147 (1998) · Zbl 0899.65008
[40] Lu, J.C.: An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method. Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts (2005)
[41] Mattsson, K.; Svärd, M.; Nordström, J., Stable and accurate artificial dissipation, J. Sci. Comput., 21, 1, 57-79 (2004) · Zbl 1085.76050
[42] Oliphant, T.E.: A Guide to NumPy, vol. 1. Trelgol Publishing USA (2006)
[43] Parsani, M.; Carpenter, Mh; Fisher, Tc; Nielsen, Ej, Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier-Stokes equations, SIAM J. Sci. Comput., 38, 5, A3129-A3162 (2016) · Zbl 1457.65149
[44] Pulliam, Th, Artificial dissipation models for the Euler equations, AIAA J., 24, 12, 1931-1940 (1986) · Zbl 0611.76075
[45] Ranocha, H.; Glaubitz, J.; Öffner, P.; Sonar, T., Stability of artificial dissipation and modal filtering for flux reconstruction schemes using summation-by-parts operators, Appl. Numer. Math., 128, 1-23 (2018) · Zbl 1404.65201
[46] Svärd, M.; Nordström, J., Review of summation-by-parts schemes for initial-boundary-value-problems, J. Comput. Phys., 268, 1, 17-38 (2014) · Zbl 1349.65336
[47] Svärd, M.; Özcan, H., Entropy-stable schemes for the Euler equations with far-field and wall boundary conditions, J. Sci. Comput., 58, 1, 61-89 (2013) · Zbl 1290.65084
[48] Tadmor, E., The numerical viscosity of entropy stable schemes for systems of conservation laws I, Math. Comput., 49, 179, 91-103 (1987) · Zbl 0641.65068
[49] Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta Numer., 12, 451-512 (2003) · Zbl 1046.65078
[50] Van Der Walt, S.; Colbert, Sc; Varoquaux, G., The NumPy array: a structure for efficient numerical computation, Comput. Sci. Eng., 13, 2, 22 (2011)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.