×

zbMATH — the first resource for mathematics

An entropy stable \(h/p\) non-conforming discontinuous Galerkin method with the summation-by-parts property. (English) Zbl 1407.65185
Summary: This work presents an entropy stable discontinuous Galerkin (DG) spectral element approximation for systems of nonlinear conservation laws with general geometric \((h)\) and polynomial order \((p)\) non-conforming rectangular meshes. The crux of the proofs presented is that the nodal DG method is constructed with the collocated Legendre-Gauss-Lobatto nodes. This choice ensures that the derivative/mass matrix pair is a summation-by-parts (SBP) operator such that entropy stability proofs from the continuous analysis are discretely mimicked. Special attention is given to the coupling between non-conforming elements as we demonstrate that the standard mortar approach for DG methods does not guarantee entropy stability for nonlinear problems, which can lead to instabilities. As such, we describe a precise procedure and modify the mortar method to guarantee entropy stability for general nonlinear hyperbolic systems on \(h/p\) non-conforming meshes. We verify the high-order accuracy and the entropy conservation/stability of fully non-conforming approximation with numerical examples.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin 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
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bohm, M., Winters, A.R., Derigs, D, Gassner, G.J., Walch, S., Saur, J.: An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations: continuous analysis and GLM divergence cleaning. Comput. Fluids (submitted), ArXiv e-prints: arXiv:1711.05576 (2017)
[2] Bui-Thanh, T.; Ghattas, O., Analysis of an \(hp\)-nonconforming discontinuous Galerkin spectral element method for wave propagation, SIAM J. Numer. Anal., 50, 1801-1826, (2012) · Zbl 1250.65120
[3] 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, b835-b867, (2014) · Zbl 1457.65140
[4] Carpenter, MH; Gottlieb, D., Spectral methods on arbitrary grids, J. Comput. Phys., 129, 74-86, (1996) · Zbl 0862.65054
[5] Carpenter, M.H., Kennedy, C.A.: Fourth-order \(2{N}\)-storage Runge-Kutta schemes. Technical report, NASA Langley Research Center (1994)
[6] Carpenter, M.H., Parsani, M., Nielsen, E.J., Fisher, T.C.: Towards an entropy stable spectral element framework for computational fluid dynamics. In: 54th AIAA Aerospace Sciences Meeting, AIAA, vol. 1058 (2016)
[7] Chandrashekar, P., Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations, Commun. Comput. Phys., 14, 1252-1286, (2013) · Zbl 1373.76121
[8] Chen, T.; Shu, C-W, Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations, J. Comput. Phys., 345, 427-461, (2016) · Zbl 1380.65253
[9] Rey Fernández, DC; Boom, PD; Zingg, DW, A generalized framework for nodal first derivative summation-by-parts operators, J. Comput. Phys., 266, 214-239, (2014) · Zbl 1311.65002
[10] Evans, L.C.: Partial Differential Equations. American Mathematical Society, New York (2012)
[11] 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
[12] Fisher, TC; Carpenter, MH; Nordström, J.; Yamaleev, NK, 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
[13] Fjordholm, US; Mishra, S.; Tadmor, E., Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography, J. Comput. Phys., 230, 5587-5609, (2011) · Zbl 1452.35149
[14] Friedrich, Lucas; Del Rey Fernández, David C.; Winters, Andrew R.; Gassner, Gregor J.; Zingg, David W.; Hicken, Jason, Conservative and Stable Degree Preserving SBP Operators for Non-conforming Meshes, Journal of Scientific Computing, 75, 657-686, (2017) · Zbl 1404.65087
[15] Gassner, GJ, A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods, SIAM J. Sci. Comput., 35, a1233-a1253, (2013) · Zbl 1275.65065
[16] Gassner, Gregor J.; Winters, Andrew R.; Hindenlang, Florian J.; Kopriva, David A., The BR1 Scheme is Stable for the Compressible Navier-Stokes Equations, Journal of Scientific Computing, 77, 154-200, (2018) · Zbl 1407.65189
[17] Gassner, GJ; Winters, AR; Kopriva, DA, Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations, J. Comput. Phys., 327, 39-66, (2016) · Zbl 1422.65280
[18] Hindenlang, F.; Gassner, GJ; Altmann, C.; Beck, A.; Staudenmaier, M.; Munz, C-D, Explicit discontinuous Galerkin methods for unsteady problems, Comput. Fluids, 61, 86-93, (2012) · Zbl 1365.76117
[19] Ismail, F.; Roe, PL, Affordable, entropy-consistent Euler flux functions II: entropy production at shocks, J. Comput. Phys., 228, 5410-5436, (2009) · Zbl 1280.76015
[20] Kopriva, DA, A conservative staggered-grid Chebyshev multidomain method for compressible flows. II. A semi-strictured method, J. Comput. Phys., 128, 475-488, (1996) · Zbl 0866.76064
[21] Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations. Scientific Computation. Springer, Berlin (2009) · Zbl 1172.65001
[22] Kopriva, DA; Woodruff, SL; Hussaini, MY, Computation of electomagnetic scattering with a non-conforming discontinuous spectral element method, Int. J. Numer. Meth. Eng., 53, 105-122, (2002) · Zbl 0994.78020
[23] Kozdon, JE; Wilcox, LC, Stable coupling of nonconforming, high-order finite difference methods, SIAM J. Sci. Comput., 3, a923-a952, (2016) · Zbl 1380.65160
[24] Mattsson, K.; Carpenter, MH, Stable and accurate interpolation operators for high-order multiblock finite difference methods, SIAM J. Sci. Comput., 32, 2298-2320, (2010) · Zbl 1216.65107
[25] Nordström, J., Lundquist, T.: On the suboptimal accuracy of summation-by-parts schemes with non-conforming block interfaces. Technical report, Linköpings Universitet (2015)
[26] 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, a3129-a3162, (2016) · Zbl 1457.65149
[27] Parsani, M.; Carpenter, MH; Nielsen, EJ, Entropy stable discontinuous interfaces coupling for the three-dimensional compressible Navier-Stokes equations, J. Comput. Phys., 290, 132-138, (2015) · Zbl 1349.76250
[28] Ray, D.; Chandrashekar, P., An entropy stable finite volume scheme for the two dimensional Navier-Stokes equations on triangular grids, Appl. Math. Comput., 314, 257-286, (2017)
[29] Sjögreen, B., Yee, H.C., Kotov, D.: Skew-symmetric splitting and stability of high order central schemes. In: Journal of Physics: Conference Series, vol. 837, p. 012019 (2017)
[30] Tadmor, E., Skew-selfadjoint form for systems of conservation laws, J. Math. Anal. Appl., 103, 428-442, (1984) · Zbl 0599.35102
[31] Tadmor, E., Entropy functions for symmetric systems of conservation laws, J. Math. Anal. Appl., 122, 355-359, (1987) · Zbl 0624.35057
[32] 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
[33] Wintermeyer, N.; Winters, AR; Gassner, GJ; Kopriva, DA, An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry, J. Comput. Phys., 340, 200-242, (2017) · Zbl 1380.65291
[34] Winters, A.R., Moura, R.C., Mengaldo, G., Gassner, G.J., Walch, S., Peiro, J., Sherwin, S.J.: A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. J. Comput. Phys. (submitted), ArXiv e-prints arXiv:1711.10180 (2017)
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.