Entropy stabilization and property-preserving limiters for ℙ₁ discontinuous Galerkin discretizations of scalar hyperbolic problems

References

[1] R. Abgrall, Essentially non-oscillatory residual distribution schemes for hyperbolic problems. J. Comput. Phys. 214 (2006), 773–808.10.1016/j.jcp.2005.10.034Suche in Google Scholar

[2] R. Abgrall, A general framework to construct schemes satisfying additional conservation relations. Application to entropy conservative and entropy dissipative schemes. J. Comput. Phys. 372 (2018), 640–666.10.1016/j.jcp.2018.06.031Suche in Google Scholar

[3] R. Abgrall, P. Öffner, and H. Ranocha, Reinterpretation and extension of entropy correction terms for residual distribution and discontinuous Galerkin schemes. arXiv:1908.04556v1, 2019.Suche in Google Scholar

[4] R. Anderson, V. Dobrev, Tz. Kolev, D. Kuzmin, and M. Quezada de Luna, R. Rieben, and V. Tomov, High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation. J. Comput. Phys. 334 (2017), 102–124.10.1016/j.jcp.2016.12.031Suche in Google Scholar

[5] S. Badia, J. Bonilla, and A. Hierro, Differentiable monotonicity-preserving schemes for discontinuous Galerkin methods on arbitrary meshes. Computer Methods Appl. Mech. Engrg. 320 (2017), 582–605.10.1016/j.cma.2017.03.032Suche in Google Scholar

[6] G. Barrenechea, V. John, and P. Knobloch, Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54 (2016) 2427–2451.10.1137/15M1018216Suche in Google Scholar

[7] T. Barth and D. C. Jespersen, The design and application of upwind schemes on unstructured meshes. AIAA Paper, (1989), 89-0366.10.2514/6.1989-366Suche in Google Scholar

[8] H. Burchard, E. Deleersnijder, and A. Meister, A higher-order conservative Patankar-type discretization for stiff systems of production–destruction equations. Appl. Numer. Math. 47 (2003), 1–30.10.1016/S0168-9274(03)00101-6Suche in Google Scholar

[9] M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, Entropy stable spectral collocation schemes for the Navier–Stokes equations: Discontinuous interfaces. SIAM J. Sci. Comput. 36 (2014), B835–B867.10.1137/130932193Suche in Google Scholar

[10] T. Chen and C. W. Shu, Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345 (2017), 427–461.10.1016/j.jcp.2017.05.025Suche in Google Scholar

[11] T. Chen and C. W. Shu, Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes. CSIAM Trans. Appl. Math. 1 (2020), 1–52.10.4208/csiam-am.2020-0003Suche in Google Scholar

[12] I. Christov and B. Popov, New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws. J. Comput. Phys. 227 (2008), No. 11, 5736–5757.10.1016/j.jcp.2008.02.007Suche in Google Scholar

[13] T. C. Fisher and M. H. Carpenter, High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. J. Comput. Phys. 252 (2013), 518–557.10.1016/j.jcp.2013.06.014Suche in Google Scholar

[14] A. Giuliani and L. Krivodonova, A moment limiter for the discontinuous Galerkin method on unstructured triangular meshes. SIAM J. Sci. Comput. 41 (2019), A508–A537.10.1137/17M1159038Suche in Google Scholar

[15] J. Glaubitz and P. Oeffner, Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered points. Appl. Numer. Math. 151 (2020), 98–118.10.1016/j.apnum.2019.12.020Suche in Google Scholar

[16] S. Gottlieb, C.-W. Shu, and E. Tadmor, Strong stability-preserving high-order time discretization methods. SIAM Review 43 (2001), 89–112.10.1137/S003614450036757XSuche in Google Scholar

[17] J.-L. Guermond and M. Nazarov, A maximum-principle preserving C⁰ finite element method for scalar conservation equations. Comput. Methods Appl. Mech. Engrg. 272 (2014), 198–213.10.1016/j.cma.2013.12.015Suche in Google Scholar

[18] J.-L. Guermond, M. Nazarov, B. Popov, and I. Tomas, Second-order invariant domain preserving approximation of the Euler equations using convex limiting. SIAM J. Sci. Computing 40 (2018), A3211-A3239.10.1137/17M1149961Suche in Google Scholar

[19] J.-L. Guermond, M. Nazarov, B. Popov, and Y. Yang, A second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations. SIAM J. Numer. Anal. 52 (2014), 2163–2182.10.1137/130950240Suche in Google Scholar

[20] J.-L. Guermond, M. Nazarov, and I. Tomas, Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems. Comput. Methods Appl. Mech. Engrg. 347 (2019), 143–175.10.1016/j.cma.2018.11.036Suche in Google Scholar

[21] J.-L. Guermond and B. Popov, Invariant domains and first-order continuous finite element approximation for hyperbolic systems. SIAM J. Numer. Anal. 54 (2016), 2466–2489.10.1137/16M1074291Suche in Google Scholar

[22] J.-L. Guermond and B. Popov, Invariant domains and second-order continuous finite element approximation for scalar conservation equations. SIAM J. Numer. Anal. 55 (2017), 3120–3146.10.1137/16M1106560Suche in Google Scholar

[23] A. Harten, On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49 (1983), 151–164.10.1016/0021-9991(83)90118-3Suche in Google Scholar

[24] H. Hajduk, D. Kuzmin, Tz. Kolev, and R. Abgrall, Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations. Comput. Methods Appl. Mech. Engrg. 359 (2020), 112658.10.1016/j.cma.2019.112658Suche in Google Scholar

[25] Y. Jiang and H. Liu, Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations. J. Comput. Appl. Math. 373 (2018), 385–409.10.1016/j.jcp.2018.03.004Suche in Google Scholar

[26] L. Krivodonova, J. Xin, J.-F. Remacle, N. Chevaugeon, and J. E. Flaherty, Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws. Appl. Numer. Math. 48 (2004), 323–338.10.1016/j.apnum.2003.11.002Suche in Google Scholar

[27] A. Kurganov, G. Petrova, and B. Popov, Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws. SIAM J. Sci. Comput. 29 (2007), 2381–2401.10.1137/040614189Suche in Google Scholar

[28] D. Kuzmin, A vertex-based hierarchical slope limiter for p-adaptive discontinuous Galerkin methods. J. Comput. Appl. Math. 233 (2010), 3077–3085.10.1016/j.cam.2009.05.028Suche in Google Scholar

[29] D. Kuzmin, Slope limiting for discontinuous Galerkin approximations with a possibly non-orthogonal Taylor basis. Int. J. Numer. Methods Fluids 71 (2013), 1178–1190.10.1002/fld.3707Suche in Google Scholar

[30] D. Kuzmin, Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws. Comput. Methods Appl. Mech. Engrg. 361 (2020), 112804.10.1016/j.cma.2019.112804Suche in Google Scholar

[31] D. Kuzmin, Algebraic flux correction I. Scalar conservation laws. In: Flux-Corrected Transport: Principles, Algorithms, and Applications (Eds. D. Kuzmin, R. Löhner, and S. Turek), Springer, 2nd edition, 2012, pp. 145–192.10.1007/978-94-007-4038-9_6Suche in Google Scholar

[32] D. Kuzmin, S. Basting, and J. N. Shadid, Linearity-preserving monotone local projection stabilization schemes for continuous finite elements. Comput. Methods Appl. Mech. Engrg. 322 (2017), 23–41.10.1016/j.cma.2017.04.030Suche in Google Scholar

[33] D. Kuzmin, H. Hajduk, and A. Rupp, Locally bound-preserving enriched Galerkin methods for the linear advection equation. Computers and Fluids 205 (2020), 104525.10.1016/j.compfluid.2020.104525Suche in Google Scholar

[34] D. Kuzmin and M. Quezada de Luna, Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws. Comput. Methods Appl. Mech. Engrg. 372 (2020), 113370.10.1016/j.cma.2020.113370Suche in Google Scholar

[35] D. Kuzmin and M. Quezada de Luna, Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws. Comput. Fluids 213 (2020), 104742.10.1016/j.compfluid.2020.104742Suche in Google Scholar

[36] C. Lohmann, Physics-Compatible Finite Element Methods for Scalar and Tensorial Advection Problems. Springer, Spektrum, 2019.10.1007/978-3-658-27737-6Suche in Google Scholar

[37] H. Luo, J. D. Baum, and R. Löhner, A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids. J. Comput. Phys. 227 (2008), 8875–8893.10.1016/j.jcp.2008.06.035Suche in Google Scholar

[38] S. A. Moe, J. A. Rossmanith, and D. C. Seal, Positivity-preserving discontinuous Galerkin methods with Lax–Wendroff time discretizations. J. Sci. Comput. 71 (2017), 44–70.10.1007/s10915-016-0291-9Suche in Google Scholar

[39] S. V. Patankar, Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York, 1980.Suche in Google Scholar

[40] W. Pazner and P.-O. Persson, Analysis and entropy stability of the line-based discontinuous Galerkin method. J. Sci. Comp. 80 (2019), 376–402.10.1007/s10915-019-00942-1Suche in Google Scholar

[41] P.-O. Persson and J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods. In: 44th AIAA Aerospace Sciences Meeting and Exhibit, 2006, p. 112.10.2514/6.2006-112Suche in Google Scholar

[42] H. Ranocha, M. Sayyari, L. Dalcin, M. Parsani, and D. I. Ketcheson, Relaxation Runge–Kutta methods: Fully-discrete explicit entropy-stable schemes for the compressible Euler and Navier–Stokes equations. SIAM J. Sci. Comput. 42 (2020), A612–A638.10.1137/19M1263480Suche in Google Scholar

[43] D. Ray, P. Chandrashekar, U. S. Fjordholm, and S. Mishra, Entropy stable scheme on two-dimensional unstructured grids for Euler equations. Commun. Comput. Phys. 19 (2016), 1111–1140.10.4208/cicp.scpde14.43sSuche in Google Scholar

[44] E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws I. Math. Comp. 49 (1987), 91–103.10.1090/S0025-5718-1987-0890255-3Suche in Google Scholar

[45] E. Tadmor, Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica 12 (2003) 451–512.10.1017/CBO9780511550157.007Suche in Google Scholar

[46] E. Tadmor, Entropy stable schemes. In: Handbook of Numerical Analysis, Vol. 17, Elsevier, 2016, pp.467–493.10.1016/bs.hna.2016.09.006Suche in Google Scholar

[47] X. Zhang and C-W. Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229 (2010), 8918–8934.10.1016/j.jcp.2010.08.016Suche in Google Scholar

[48] X. Zhang and C-W. Shu, Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A 467 (2011), 2752–2776.10.1098/rspa.2011.0153Suche in Google Scholar