{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,23]],"date-time":"2026-03-23T10:41:50Z","timestamp":1774262510998,"version":"3.50.1"},"reference-count":66,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,3,1]],"date-time":"2020-03-01T00:00:00Z","timestamp":1583020800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,3,3]],"date-time":"2020-03-03T00:00:00Z","timestamp":1583193600000},"content-version":"vor","delay-in-days":2,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"crossref","award":["714487"],"award-info":[{"award-number":["714487"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2020,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This work is focused on the entropy analysis of a semi-discrete nodal discontinuous Galerkin spectral element method (DGSEM) on moving meshes for hyperbolic conservation laws. The DGSEM is constructed with a local tensor-product Lagrange-polynomial basis computed from Legendre\u2013Gauss\u2013Lobatto points. Furthermore, the collocation of interpolation and quadrature nodes is used in the spatial discretization. This approach leads to discrete derivative approximations in space that are summation-by-parts (SBP) operators. On a static mesh, the SBP property and suitable two-point flux functions, which satisfy the entropy condition from Tadmor, allow to mimic results from the continuous entropy analysis, if it is ensured that properties such as positivity preservation (of the water height, density or pressure) are satisfied on the discrete level. In this paper, Tadmor\u2019s condition is extended to the moving mesh framework. We show that the volume terms in the semi-discrete moving mesh DGSEM do not contribute to the discrete entropy evolution when a two-point flux function that satisfies the moving mesh entropy condition is applied in the split form DG framework. The discrete entropy behavior then depends solely on the interface contributions and on the domain boundary contribution. The interface contributions are directly controlled by proper choice of the numerical element interface fluxes. If an entropy conserving two-point flux is chosen, the interface contributions vanish. To increase the robustness of the discretization we use so-called entropy stable two-point fluxes at the interfaces that are guaranteed entropy dissipative and thus give a bound on the interface contributions in the discrete entropy balance. The remaining boundary condition contributions depend on the type of the considered boundary condition. E.g. for periodic boundary conditions that are of entropy conserving type, our methodology with the entropy conserving interface fluxes is fully entropy conservative and with the entropy stable interface fluxes is guaranteed entropy stable. The presented proof does not require any exactness of quadrature in the spatial integrals of the variational forms. As it is the case for static meshes, these results rely on the assumption that additional properties like positivity preservation are satisfied on the discrete level. Besides the entropy stability, the time discretization of the moving mesh DGSEM will be investigated and it will be proven that the moving mesh DGSEM satisfies the free stream preservation property for an arbitrary <jats:italic>s<\/jats:italic>-stage Runge\u2013Kutta method, when periodic boundary conditions are used. The theoretical properties of the moving mesh DGSEM will be validated by numerical experiments for the compressible Euler equations with periodic boundary conditions.<\/jats:p>","DOI":"10.1007\/s10915-020-01171-7","type":"journal-article","created":{"date-parts":[[2020,3,3]],"date-time":"2020-03-03T06:03:26Z","timestamp":1583215406000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Entropy Stable Discontinuous Galerkin Schemes on Moving Meshes for Hyperbolic Conservation Laws"],"prefix":"10.1007","volume":"82","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6769-9018","authenticated-orcid":false,"given":"Gero","family":"Schn\u00fccke","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nico","family":"Krais","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Thomas","family":"Bolemann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gregor J.","family":"Gassner","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,3,3]]},"reference":[{"key":"1171_CR1","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1016\/j.jcp.2013.12.019","volume":"260","author":"Y Abe","year":"2014","unstructured":"Abe, Y., Nonomura, T., Iizuka, N., Fujii, K.: Geometric interpretations and spatial symmetry property of metrics in the conservative form for high-order finite-difference schemes on moving and deforming grids. J. Comput. Phys. 260, 163\u2013203 (2014)","journal-title":"J. Comput. Phys."},{"key":"1171_CR2","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/978-3-642-58535-7_5","volume-title":"An Introduction to Recent Developments in Theory and Numerics for Conservation Laws","author":"TJ Barth","year":"1999","unstructured":"Barth, T.J.: Numerical methods for gasdynamic systems on unstructured meshes. In: Kr\u00f6ner, D., Ohlberger, M., Rohde, C. (eds.) An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, pp. 195\u2013285. Springer, Berlin (1999)"},{"key":"1171_CR3","volume-title":"Introduction to Matrix Analysis, volume 19 of Classics in Applied Mathematics","author":"R Bellman","year":"1987","unstructured":"Bellman, R.: Introduction to Matrix Analysis, volume 19 of Classics in Applied Mathematics, 2nd edn. SIAM, Philadelphia (1987)","edition":"2"},{"key":"1171_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.06.027","author":"M Bohm","year":"2018","unstructured":"Bohm, M., Winters, A.R., Gassner, G.J., Derigs, D., Hindenlang, F., Saur, J.: An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: theory and numerical verification. J. Comput. Phys. (2018). https:\/\/doi.org\/10.1016\/j.jcp.2018.06.027","journal-title":"J. Comput. Phys."},{"key":"1171_CR5","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1016\/j.jcp.2017.06.022","volume":"346","author":"W Boscheri","year":"2017","unstructured":"Boscheri, W., Dumbser, M.: Arbitrary-Lagrangian\u2013Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. J. Comput. Phys. 346, 449\u2013479 (2017)","journal-title":"J. Comput. Phys."},{"key":"1171_CR6","doi-asserted-by":"publisher","first-page":"346","DOI":"10.1016\/j.jcp.2018.02.033","volume":"362","author":"J Chan","year":"2018","unstructured":"Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 362, 346\u2013374 (2018)","journal-title":"J. Comput. Phys."},{"key":"1171_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-30726-6","volume-title":"Spectral Methods: Fundamentals in Single Domains","author":"C Canuto","year":"2006","unstructured":"Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2006)"},{"key":"1171_CR8","doi-asserted-by":"publisher","first-page":"1252","DOI":"10.4208\/cicp.170712.010313a","volume":"14","author":"P Chandrashekar","year":"2013","unstructured":"Chandrashekar, P.: Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier\u2013Stokes equations. Commun. Comput. Phys. 14, 1252\u20131286 (2013)","journal-title":"Commun. Comput. Phys."},{"key":"1171_CR9","unstructured":"Chavent, G., Cockburn, B.: The local projection $$P^0P^1$$-discontinuous Galerkin finite element method for scalar conservation laws. ESAIM Math. Model. Num. ($$M^2AN$$) 23, 565\u2013592 (1989)"},{"key":"1171_CR10","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1016\/j.jcp.2017.05.025","volume":"345","author":"T Chen","year":"2017","unstructured":"Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427\u2013461 (2017)","journal-title":"J. Comput. Phys."},{"key":"1171_CR11","doi-asserted-by":"publisher","first-page":"1157","DOI":"10.1002\/cpa.21537","volume":"68","author":"E Chiodaroli","year":"2015","unstructured":"Chiodaroli, E., De Lellis, C., Kreml, O.: Global Ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Math. 68, 1157\u20131190 (2015)","journal-title":"Commun. Pure Appl. Math."},{"key":"1171_CR12","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1023\/A:1012873910884","volume":"16","author":"B Cockburn","year":"2001","unstructured":"Cockburn, B., Shu, C.-W.: Runge\u2013Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173\u2013261 (2001)","journal-title":"J. Sci. Comput."},{"key":"1171_CR13","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1016\/j.jcp.2017.12.015","volume":"356","author":"J Crean","year":"2018","unstructured":"Crean, J., Hicken, J.E., Fern\u00e1ndez, D.C.D.R., Zingg, D.W., Carpenter, M.H.: Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. J. Comput. Phys. 356, 410\u2013438 (2018)","journal-title":"J. Comput. Phys."},{"key":"1171_CR14","doi-asserted-by":"publisher","first-page":"108775","DOI":"10.1016\/j.jcp.2019.06.051","volume":"397","author":"L Dalcin","year":"2019","unstructured":"Dalcin, L., Rojas, D., Zampini, S., Fern\u00e1ndez, D.C.D.R., Carpenter, M.H., Parsani, M.: Conservative and entropy stable solid wall boundary conditions for the compressible Navier\u2013Stokes equations: adiabatic wall and heat entropy transfer. J. Comput. Phys. 397, 108775 (2019)","journal-title":"J. Comput. Phys."},{"key":"1171_CR15","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1007\/BF00251724","volume":"82","author":"RJ Di Perna","year":"1983","unstructured":"Di Perna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82, 27\u201370 (1983)","journal-title":"Arch. Ration. Mech. Anal."},{"key":"1171_CR16","volume-title":"Mathematical Aspects of Discontinuous Galerkin Methods","author":"DA Di Pietro","year":"2011","unstructured":"Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69. Springer, Berlin (2011)"},{"key":"1171_CR17","first-page":"1","volume-title":"Arbitrary Lagrangian\u2013Eulerian Methods. Encyclopedia of Computational Mechanics","author":"J Donea","year":"2017","unstructured":"Donea, J., Huerta, A., Ponthot, J.P., Rodr\u00edguez-Ferran, A.: Arbitrary Lagrangian\u2013Eulerian Methods. Encyclopedia of Computational Mechanics, 2nd edn, pp. 1\u201323. Wiley, Hoboken (2017)","edition":"2"},{"key":"1171_CR18","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1006\/jcph.2001.6932","volume":"174","author":"C Farhat","year":"2001","unstructured":"Farhat, C., Geuzaine, P., Grandmont, C.: The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids. J. Comput. Phys. 174, 669\u2013694 (2001)","journal-title":"J. Comput. Phys."},{"key":"1171_CR19","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/j.compfluid.2014.02.016","volume":"95","author":"DCDR Fern\u00e1ndez","year":"2014","unstructured":"Fern\u00e1ndez, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Comput. Fluids 95, 171\u2013196 (2014)","journal-title":"Comput. Fluids"},{"key":"1171_CR20","doi-asserted-by":"publisher","first-page":"518","DOI":"10.1016\/j.jcp.2013.06.014","volume":"252","author":"TC Fisher","year":"2013","unstructured":"Fisher, T.C., Carpenter, M.H.: High order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518\u2013557 (2013)","journal-title":"J. Comput. Phys."},{"key":"1171_CR21","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1007\/s10915-019-00933-2","volume":"80","author":"L Friedrich","year":"2019","unstructured":"Friedrich, L., Schn\u00fccke, G., Winters, A.R., Fern\u00e1ndez, D.C.D.R., Gassner, G.J., Carpenter, M.H.: Entropy stable space-time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws. J. Sci. Comput. 80, 175\u2013222 (2019)","journal-title":"J. Sci. Comput."},{"key":"1171_CR22","doi-asserted-by":"publisher","first-page":"A1233","DOI":"10.1137\/120890144","volume":"35","author":"GJ Gassner","year":"2013","unstructured":"Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35, A1233\u2013A1253 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"1171_CR23","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/j.jcp.2016.09.013","volume":"327","author":"GJ Gassner","year":"2016","unstructured":"Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327, 39\u201366 (2016)","journal-title":"J. Comput. Phys."},{"key":"1171_CR24","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-018-0692-z","volume":"77","author":"GJ Gassner","year":"2018","unstructured":"Gassner, G.J., Winters, A.R., Hindenlang, F.J., Kopriva, D.A.: The BR1 scheme is stable for the compressible Navier\u2013Stokes equations. J Sci. Comput. 77, 1\u201347 (2018)","journal-title":"J Sci. Comput."},{"key":"1171_CR25","volume-title":"Hyperbolic Systems of Conservation Laws","author":"E Godlewski","year":"1991","unstructured":"Godlewski, E., Raviart, P.-A.: Hyperbolic Systems of Conservation Laws. Ellipses, Paris (1991)"},{"key":"1171_CR26","first-page":"521","volume":"139","author":"SK Godunov","year":"1961","unstructured":"Godunov, S.K.: An interesting class of quasilinear systems. Dokl. Acad. Nauk SSSR 139, 521\u2013523 (1961)","journal-title":"Dokl. Acad. Nauk SSSR"},{"issue":"2","key":"1171_CR27","doi-asserted-by":"publisher","first-page":"A385","DOI":"10.1137\/16M1063034","volume":"39","author":"JL Guermond","year":"2017","unstructured":"Guermond, J.L., Popov, B., Saavedra, L., Yang, Y.: Invariant domains preserving arbitrary Lagrangian Eulerian approximation of hyperbolic systems with continuous finite elements. SIAM J. Sci. Comput. 39(2), A385\u2013A414 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"1171_CR28","doi-asserted-by":"publisher","first-page":"1467","DOI":"10.1016\/S0045-7825(00)00173-0","volume":"190","author":"H Guillard","year":"2000","unstructured":"Guillard, H., Farhat, C.: On the significance of the geometric conservation law for flow computations on moving meshes. Comput. Method. Appl. Mech. Eng. 190, 1467\u20131482 (2000)","journal-title":"Comput. Method. Appl. Mech. Eng."},{"key":"1171_CR29","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1016\/0021-9991(83)90118-3","volume":"49","author":"A Harten","year":"1983","unstructured":"Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49, 151\u2013164 (1983)","journal-title":"J. Comput. Phys."},{"key":"1171_CR30","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1007\/978-3-319-12886-3_8","volume-title":"IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach","author":"FJ Hindenlang","year":"2015","unstructured":"Hindenlang, F.J., Bolemann, T., Munz, C.-D.: Mesh curving techniques for high order discontinuous Galerkin simulations. In: Kroll, N., Hirsch, C., Bassi, F., Johnston, C., Hillewaert, K. (eds.) IDIHOM: Industrialization of High-Order Methods-A Top-Down Approach, pp. 133\u2013152. Springer, Cham (2015)"},{"key":"1171_CR31","unstructured":"Hindenlang, F.J., Gassner, G.J., Kopriva, D.A.: Stability of wall boundary condition procedures for discontinuous Galerkin spectral element approximations of the compressible Euler equations (2019). arXiv preprint arXiv:1901.04924"},{"key":"1171_CR32","volume-title":"Adaptive Moving Mesh Methods","author":"W Huang","year":"2010","unstructured":"Huang, W., Russell, R.D.: Adaptive Moving Mesh Methods, vol. 174. Springer, Berlin (2010)"},{"key":"1171_CR33","doi-asserted-by":"publisher","first-page":"5410","DOI":"10.1016\/j.jcp.2009.04.021","volume":"228","author":"F Ismail","year":"2009","unstructured":"Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228, 5410\u20135436 (2009)","journal-title":"J. Comput. Phys."},{"key":"1171_CR34","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/S0168-9274(99)00141-5","volume":"35","author":"CA Kennedy","year":"2000","unstructured":"Kennedy, C.A., Carpenter, M.H., Lewis, R.M.: Low-storage, explicit Runge\u2013Kutta schemes for the compressible Navier\u2013Stokes equations. Appl. Numer. Math. 35, 177\u2013219 (2000)","journal-title":"Appl. Numer. Math."},{"key":"1171_CR35","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s10915-005-9070-8","volume":"26","author":"DA Kopriva","year":"2006","unstructured":"Kopriva, D.A.: Metric identities and the discontinuous spectral element method on curvilinear meshes. J. Sci. Comput. 26, 301\u2013327 (2006)","journal-title":"J. Sci. Comput."},{"key":"1171_CR36","doi-asserted-by":"publisher","DOI":"10.1007\/978-90-481-2261-5","volume-title":"Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers","author":"DA Kopriva","year":"2009","unstructured":"Kopriva, D.A.: Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers. Springer, Berlin (2009)"},{"key":"1171_CR37","doi-asserted-by":"publisher","first-page":"148","DOI":"10.1016\/j.compfluid.2016.05.023","volume":"139","author":"DA Kopriva","year":"2016","unstructured":"Kopriva, D.A., Winters, A.R., Bohm, M., Gassner, G.J.: A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes. Comput. Fluids 139, 148\u2013160 (2016)","journal-title":"Comput. Fluids"},{"key":"1171_CR38","doi-asserted-by":"publisher","first-page":"199","DOI":"10.3402\/tellusa.v24i3.10634","volume":"24","author":"H-O Kreiss","year":"1972","unstructured":"Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199\u2013215 (1972)","journal-title":"Tellus"},{"key":"1171_CR39","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1070\/SM1970v010n02ABEH002156","volume":"10","author":"SN Kru\u017ekov","year":"1970","unstructured":"Kru\u017ekov, S.N.: First order quasilinear equations in several independent variables. Math. USSR Sbornik 10, 217\u2013243 (1970)","journal-title":"Math. USSR Sbornik"},{"key":"1171_CR40","doi-asserted-by":"publisher","first-page":"1968","DOI":"10.1137\/S003614290240069X","volume":"40","author":"PG LeFloch","year":"2002","unstructured":"LeFloch, P.G., Mercier, J.-M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40, 1968\u20131992 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"1171_CR41","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1016\/0045-7825(96)01028-6","volume":"134","author":"M Lesoinne","year":"1996","unstructured":"Lesoinne, M., Farhat, C.: Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Comput. Method. Appl. Mech. Eng. 134, 71\u201390 (1996)","journal-title":"Comput. Method. Appl. Mech. Eng."},{"key":"1171_CR42","doi-asserted-by":"publisher","first-page":"1030","DOI":"10.2514\/3.61273","volume":"17","author":"CK Lombard","year":"1979","unstructured":"Lombard, C.K., Thomas, P.D.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17, 1030\u20131037 (1979)","journal-title":"AIAA J."},{"key":"1171_CR43","doi-asserted-by":"publisher","first-page":"128","DOI":"10.1006\/jcph.1999.6331","volume":"155","author":"I Lomtev","year":"1999","unstructured":"Lomtev, I., Kirby, R.M., Karniadakis, G.E.: A discontinuous Galerkin ALE method for compressible viscous flows in moving domains. J. Comput. Phys. 155, 128\u2013159 (1999)","journal-title":"J. Comput. Phys."},{"key":"1171_CR44","doi-asserted-by":"publisher","first-page":"4724","DOI":"10.1016\/j.jcp.2010.03.011","volume":"229","author":"R Loub\u00e8re","year":"2010","unstructured":"Loub\u00e8re, R., Maire, P.H., Shashkov, M., Breil, J., Galera, S.: ReALE: a reconnection-based arbitrary-Lagrangian\u2013Eulerian method. J. Comput. Phys. 229, 4724\u20134761 (2010)","journal-title":"J. Comput. Phys."},{"key":"1171_CR45","doi-asserted-by":"publisher","first-page":"1750","DOI":"10.1093\/mnras\/stt2003","volume":"437","author":"F Marinacci","year":"2013","unstructured":"Marinacci, F., Pakmor, R., Springel, V.: The formation of disc galaxies in high-resolution moving-mesh cosmological simulations. Mon. Not. R. Astron. Soc. 437, 1750\u20131775 (2013)","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"1171_CR46","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1016\/j.jcp.2005.08.018","volume":"213","author":"DJ Mavriplis","year":"2006","unstructured":"Mavriplis, D.J., Yang, Z.: Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes. J. Comput. Phys. 213, 557\u2013573 (2006)","journal-title":"J. Comput. Phys."},{"key":"1171_CR47","doi-asserted-by":"crossref","unstructured":"Merriam, M.L.: Towards a rigorous approach to artificial dissipation. In: Computational Fluid Dynamics. No. AIAA-Paper-89-0471; CONF-890123. National Aeronautics and Space Administration, Moffett Field, CA, USA. Ames Research Center (1989)","DOI":"10.2514\/6.1989-471"},{"key":"1171_CR48","doi-asserted-by":"publisher","first-page":"1876","DOI":"10.1016\/j.jcp.2010.11.038","volume":"230","author":"CAA Minoli","year":"2011","unstructured":"Minoli, C.A.A., Kopriva, D.A.: Discontinuous Galerkin spectral element approximations on moving meshes. J. Comput. Phys. 230, 1876\u20131902 (2011)","journal-title":"J. Comput. Phys."},{"key":"1171_CR49","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1016\/0022-0396(80)90089-3","volume":"37","author":"MS Mock","year":"1980","unstructured":"Mock, M.S.: Systems of conservation laws of mixed type. J. Differ. Equ. 37, 70\u201388 (1980)","journal-title":"J. Differ. Equ."},{"key":"1171_CR50","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1016\/j.jcp.2016.10.056","volume":"330","author":"RC Moura","year":"2017","unstructured":"Moura, R.C., Mengaldo, G., Peir\u00f3, J., Sherwin, S.J.: On the eddy-resolving capability of high-order discontinuous Galerkin approaches to implicit LES\/under-resolved DNS of Euler turbulence. J. Comput. Phys. 330, 615\u2013623 (2017)","journal-title":"J. Comput. Phys."},{"key":"1171_CR51","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1016\/j.jcp.2015.02.027","volume":"291","author":"S Nikkar","year":"2015","unstructured":"Nikkar, S., Nordstr\u00f6m, J.: Fully discrete energy stable high order finite difference methods for hyperbolic problems in deforming domains. J. Comput. Phys. 291, 82\u201398 (2015)","journal-title":"J. Comput. Phys."},{"key":"1171_CR52","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/j.jfluidstructs.2009.11.002","volume":"26","author":"VT Nguyen","year":"2010","unstructured":"Nguyen, V.T.: An arbitrary Lagrangian\u2013Eulerian discontinuous Galerkin method for simulations of flows over variable geometries. J. Fluid. Struct. 26, 312\u2013329 (2010)","journal-title":"J. Fluid. Struct."},{"key":"1171_CR53","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1016\/j.jcp.2015.03.026","volume":"292","author":"M Parsani","year":"2015","unstructured":"Parsani, M., Carpenter, M.H., Nielsen, E.J.: Entropy stable wall boundary conditions for the three-dimensional compressible Navier\u2013Stokes equations. J. Comput. Phys. 292, 88\u2013113 (2015)","journal-title":"J. Comput. Phys."},{"key":"1171_CR54","doi-asserted-by":"publisher","first-page":"1585","DOI":"10.1016\/j.cma.2009.01.012","volume":"198","author":"PO Persson","year":"2009","unstructured":"Persson, P.O., Bonet, J., Peraire, J.: Discontinuous Galerkin solution of the Navier\u2013Stokes equations on deformable domains. Comput. Methods Appl. Mech. Eng. 198, 1585\u20131595 (2009)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1171_CR55","volume-title":"Generalised Summation-by-Parts Operators and Entropy Stability of Numerical Methods for Hyperbolic Balance Laws","author":"H Ranocha","year":"2018","unstructured":"Ranocha, H.: Generalised Summation-by-Parts Operators and Entropy Stability of Numerical Methods for Hyperbolic Balance Laws. Cuvillier, G\u00f6ttingen (2018)"},{"key":"1171_CR56","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-004-5407-y","volume":"24","author":"CW Shu","year":"2005","unstructured":"Shu, C.W., Don, W.S., Gottlieb, D., Schilling, O., Jameson, L.: Numerical convergence study of nearly incompressible, inviscid Taylor\u2013Green vortex flow. J. Sci. Comput. 24, 1\u201327 (2005)","journal-title":"J. Sci. Comput."},{"key":"1171_CR57","doi-asserted-by":"publisher","first-page":"791","DOI":"10.1111\/j.1365-2966.2009.15715.x","volume":"401","author":"V Springel","year":"2010","unstructured":"Springel, V.: E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh. Mon. Not. R. Astron. Soc. 401, 791\u2013851 (2010)","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"1171_CR58","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1090\/S0025-5718-1987-0890255-3","volume":"49","author":"Eitan Tadmor","year":"1987","unstructured":"Tadmor, Eitan: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49, 91\u2013103 (1987)","journal-title":"Math. Comput."},{"key":"1171_CR59","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1017\/S0962492902000156","volume":"12","author":"E Tadmor","year":"2003","unstructured":"Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451\u2013512 (2003)","journal-title":"Acta Numer."},{"key":"1171_CR60","doi-asserted-by":"crossref","unstructured":"Wang, L., Persson, P.O.: High-order discontinuous Galerkin simulations on moving domains using ALE formulations and local remeshing and projections. In: 53rd AIAA Aerospace Sciences Meeting. AIAA SciTech Forum (AIAA 2015-0820). https:\/\/doi.org\/10.2514\/6.2015-0820 . Cited 20 Feb 2020","DOI":"10.2514\/6.2015-0820"},{"key":"1171_CR61","doi-asserted-by":"publisher","first-page":"200","DOI":"10.1016\/j.jcp.2017.03.036","volume":"340","author":"N Wintermeyer","year":"2017","unstructured":"Wintermeyer, N., Winters, A.R., Gassner, G.J., Kopriva, D.A.: 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\u2013242 (2017)","journal-title":"J. Comput. Phys."},{"key":"1171_CR62","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jcp.2018.06.016","volume":"372","author":"AR Winters","year":"2018","unstructured":"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. 372, 1\u201321 (2018)","journal-title":"J. Comput. Phys."},{"key":"1171_CR63","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1016\/j.jcp.2016.12.006","volume":"332","author":"AR Winters","year":"2017","unstructured":"Winters, A.R., Derigs, D., Gassner, G.J., Walch, S.: A uniquely defined entropy stable matrix dissipation operator for high Mach number ideal MHD and compressible Euler simulations. J. Comput. Phys. 332, 274\u2013289 (2017)","journal-title":"J. Comput. Phys."},{"key":"1171_CR64","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1016\/j.jcp.2014.01.022","volume":"263","author":"AR Winters","year":"2014","unstructured":"Winters, A.R., Kopriva, D.A.: ALE-DGSEM approximation of wave reflection and transmission from a moving medium. J. Comput. Phys. 263, 233\u2013267 (2014)","journal-title":"J. Comput. Phys."},{"key":"1171_CR65","unstructured":"Winters, A.R.: Discontinuous Galerkin spectral element approximations for the reflection and transmission of waves from moving material interfaces. Dissertation, The Florida State University (2014)"},{"key":"1171_CR66","doi-asserted-by":"publisher","first-page":"108897","DOI":"10.1016\/j.jcp.2019.108897","volume":"399","author":"NK Yamaleev","year":"2019","unstructured":"Yamaleev, N.K., Fernandez, D.C.D.R., Lou, J., Carpenter, M.H.: Entropy stable spectral collocation schemes for the 3-D Navier\u2013Stokes equations on dynamic unstructured grids. J. Comput. Phys. 399, 108897 (2019)","journal-title":"J. Comput. Phys."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01171-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-020-01171-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-020-01171-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,3,3]],"date-time":"2021-03-03T00:23:15Z","timestamp":1614730995000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-020-01171-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,3]]},"references-count":66,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2020,3]]}},"alternative-id":["1171"],"URL":"https:\/\/doi.org\/10.1007\/s10915-020-01171-7","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,3]]},"assertion":[{"value":"21 December 2018","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 February 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 March 2020","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"69"}}