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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
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##### 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)
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