On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems

Abstract This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first hyperbolic reformulations nonlinear dispersive systems based an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We consider the two different systems, namely Serre–Green–Naghdi model water waves defocusing Schrödinger equation. The reformulation equation endowed curl involution constraint that needs be properly accounted for in multiple space dimensions. show original proposed (2018) only weakly multi-dimensional case strong hyperbolicity can restored at aid novel thermodynamically compatible GLM cleaning accounts PDE system. one two-dimensional numerical results both compare them available exact, experimental reference solutions whenever possible.