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

Saray Busto,Michael Dumbser,Cipriano Escalante,Nicolas Favrie,Sergey Gavrilyuk

doi:10.1007/s10915-021-01429-8

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 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.

Highlights

Nonlinear dispersive systems can be found in many different areas of computational mechanics, ranging from large scale dispersive free surface shallow water flows [12,28,62,87,93,94] over multi-phase flows with surface tension [13,39] down to quantum fluid mechanics [14,15,65,75]
The rest of this paper is structured as follows: in Sect. 2 we present the two nonlinear dispersive models that we want to study, namely the hyperbolic reformulation of the Serre–Green–Naghdi system of dispersive free surface water waves forwarded by Favrie and Gavrilyuk in [56] and the hyperbolic reformulation of the nonlinear defocusing Schrödinger equation proposed by Dhaouadi et al in [38]
For the numerical simulation of the hyperbolic reformulations of the nonlinear dispersive systems introduced in the previous section, and which can all be cast into the general form (44), we employ the family of high order accurate fully-discrete one-step arbitrary high order derivative (ADER)

Summary

Introduction

Nonlinear dispersive systems can be found in many different areas of computational mechanics, ranging from large scale dispersive free surface shallow water flows [12,28,62,87,93,94] over multi-phase flows with surface tension [13,39] down to quantum fluid mechanics [14,15,65,75]. More recent work on the topic regards the hyperbolic reformulation of advection-diffusion equations [80,84,85,99], of the compressible Navier-Stokes equations [17,47,88] as well as hyperbolic reformulations of nonlinear dispersive systems [3,4,26,51,52,63,64,79,90]. We would like to point out a particularity of the hyperbolic dispersive system proposed in [3], since it can be derived from the depth averaged compressible Euler equations, while most depth-averaged shallow water systems are usually derived from the governing equations of an incompressible fluid. The well-known theory of rational extended thermodynamics [82] makes use of hyperbolic relaxation systems to model the effects of higher order derivative terms

Methods

Results

Conclusion