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

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

Journal

Journal of Scientific Computing
ISSN: 0885-7474

Output data

Year: 2021, Volume: 87, Number: 2, Article number : 48, Pages count : DOI: 10.1007/s10915-021-01429-8

Tags

Curl involution constraint; High order ADER discontinous Galerkin schemes with subcell finite volume limiter; Hyperbolic reformulations of nonlinear dispersive systems; Nonlinear Schrödinger equation; Serre–Green–Naghdi model; Thermodynamically compatible GLM curl cleaning; Well-balancing

Authors

Busto S. ¹ , Dumbser M. ¹ , Escalante C. ² , Favrie N. ³ , Gavrilyuk S. ^3,4

Affiliations

1	Department of Civil, Environmental and Mechanical Engineering, University of Trento, Via Mesiano 77, Trento, 38123, Italy
2	Department of Mathematics, University of Cordoba, Córdoba, 14071, Spain
3	Aix-Marseille Univ and CNRS UMR 7343 IUSTI, 5 rue Enrico Fermi, Marseille, 13453, France
4	Lavrentyev Institute of Hydrodynamics, 15 Lavrentyev Ave., Novosibirsk, 630090, Russian Federation

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 order hyperbolic reformulations of nonlinear dispersive systems based on 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 hyperbolic reformulations of two different nonlinear dispersive systems, namely the Serre–Green–Naghdi model of dispersive water waves and the defocusing nonlinear Schrödinger equation. The first order hyperbolic reformulation of the Schrödinger equation is endowed with a curl involution constraint that needs to be properly accounted for in multiple space dimensions. We show that the original model proposed in Dhaouadi et al. (2018) is only weakly hyperbolic in the multi-dimensional case and that strong hyperbolicity can be restored at the aid of a novel thermodynamically compatible GLM curl cleaning approach that accounts for the curl involution constraint in the PDE system. We show one and two-dimensional numerical results applied to both systems and compare them with available exact, numerical and experimental reference solutions whenever possible. © 2021, The Author(s).

Busto S. , Dumbser M. , Escalante C. , Favrie N. , Gavrilyuk S.
On High Order ADER Discontinuous Galerkin Schemes for First Order Hyperbolic Reformulations of Nonlinear Dispersive Systems
Journal of Scientific Computing. 2021. V.87. N2. 48 . DOI: 10.1007/s10915-021-01429-8 WOS Scopus OpenAlex

≡ Web of science:	WOS:000632916100001
≡ Scopus:	2-s2.0-85103377766
≡ OpenAlex:	W3140478212