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 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.

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

“…This guarantees that trP ≥ 0 by construction also at the discrete level for all times, since in terms of Q the trace of the Reynolds Fig. 6 Numerical simulation of the SWASI experiment with a fourth order ADER-DG scheme at time t = 57 s applied to the model for unsteady turbulent shallow water flows ( 14)- (18). Water depth (top left), Froude number (top right), angular velocity (bottom left) and limiter map with limited cells highlighted in red and unlimited cells plotted in blue (bottom right) (Color figure online) Fig.…”

“…In the future we plan to extend the new family of thermodynamically compatible schemes to the equations of nonlinear hyperelasticity [14,67,75,77,87,102] and to the unified hyperbolic model of continuum mechanics [13,17,47,93,102], as well as to hyperbolic reformulations of dispersive systems [7,18,37,58]. Further work will also concern the extension of the discrete Godunov formalism presented in this paper to higher order semi-discrete discontinuous Galerkin finite element schemes, see e.g.…”

“…In [69] a special split scheme was developed, splitting the original system (1)-( 5) into two quasi-conservative subsystems and during the solution of each of the subsystems, energy conservation was rigorously enforced. In this paper, two new unsplit schemes have been proposed for the discretization of ( 14)- (18). In the case of the path-conservative ADER-DG scheme described in Sect.…”

“…In order to maintain a strict compatibility of the discrete energy conservation law (18) with the discrete trace of P, for the inviscid case ε = 0 we proceed as follows: at the end of each time step, we compute in each degree of freedom of the DG scheme and in each control volume of the subcell finite volume limiter the trace of P from the total energy and subsequently rescale Q according to…”

“…The severe time step restriction induced by the inclusion of higher order derivatives, [83,122,123], and nonlinear dispersive equations, [51,53,54], has driven to the development of fully implicit approaches, [44], whose major disadvantage is the solution of the resulting ill-conditioned algebraic systems. An alternative approach recently proposed is the use of hyperbolic reformulations of dispersive models which allow for more efficient discretizations, [7,8,18,52].…”