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.

本文是基于Dhaouadi等人引入的扩展拉格朗日方法，将具有子单元有限体积限制器的任意高阶全离散单步ADER不连续Galerkin方案应用于子类有限体积限制器，将其应用于一类新的非线性色散系统的一阶双曲重构。（2018年Stud Appl数学207：1–20），Favrie和Gavrilyuk（非线性30：2718–2736，2017）。我们考虑了两个不同的非线性色散系统的双曲重构，即色散水波的Serre-Green-Naghdi模型和散焦非线性Schrödinger方程。Schrödinger方程的一阶双曲线重制具有卷曲对合约束，需要在多个空间维度中适当考虑。我们证明了Dhaouadi等人提出的原始模型。（2018）仅在多维情况下是弱双曲的，并且可以通过一种新颖的热力学兼容的GLM卷曲清洁方法来恢复强双曲性，该方法解决了PDE系统中的卷曲对合约束。我们显示了应用于这两个系统的一维和二维数值结果，并在可能的情况下将其与可用的精确，数值和实验参考解决方案进行了比较。