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Provided the scheme is discretely conservative and satisfies discrete geometric conservation laws, we derive a condition on the time step to guarantee that the cell-averaged approximate solution is a convex combination of states that are contained in all the invariant domains of the conservation laws. This condition is based on the existence of a pseudo-equilibrium solution that satisfies a balance of the numerical fluxes at the faces of each element. For application purposes, we use these positive states to define local bounds which are then imposed to the high-order approximate solution within the cells via existing linear scaling limiters. The scheme is invariant domain preserving after the limiting procedure. Numerical experiments are presented with modal and nodal discontinuous Galerkin schemes and confirm the robustness and stability enhancement of the present approach.<\/jats:p>","DOI":"10.1137\/22m1492015","type":"journal-article","created":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T17:13:46Z","timestamp":1687972426000},"page":"A1385-A1412","source":"Crossref","is-referenced-by-count":9,"title":["Invariant Domain Preserving High-Order Spectral Discontinuous Approximations of Hyperbolic Systems"],"prefix":"10.1137","volume":"45","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8228-171X","authenticated-orcid":true,"given":"Valentin","family":"Carlier","sequence":"first","affiliation":[{"name":"Max-Planck Institute for Plasma Physics, Boltzmannstr. 2, D-85748 Garching, Germany."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8584-0361","authenticated-orcid":true,"given":"Florent","family":"Renac","sequence":"additional","affiliation":[{"name":"DAAA, ONERA, Universit\u00e9 Paris Saclay F-92322 Ch\u00e2tillon, France."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2023,6,23]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01584551"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2009.11.010"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1997.5454"},{"key":"ref4","first-page":"1","volume":"11","author":"Berthon C.","year":"2014","journal-title":"Int. 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