In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al. We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian AMR meshes. We then present and discuss the unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mechanics developed by Godunov, Peshkov, and Romenski (GPR model), which allows to describe fluid and solid mechanics in one single and unified first order hyperbolic system. In order to deal with free surface and moving boundary problems, a simple diffuse interface approach is employed, which is compatible with Eulerian schemes on fixed grids as well as direct Arbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples of moving boundary problems in fluid and solid mechanics.

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 onlyweakly hyperbolicin 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.

In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatibleviscous extensionof the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provablythermodynamically compatiblesemi-discrete finite volume scheme that makes direct use of theGodunov formof the equations and can therefore be called adiscrete Godunov formalism. The new method mimics the underlying continuous viscous systemexactlyat the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.