Quantum hydrodynamics with trajectories: The nonlinear conservation form mixed/discontinuous Galerkin method with applications in chemistry

Abstract

a b s t r a c tWe present a solution to the conservation form (Eulerian form) of the quantum hydrodynamic equations which arise in chemical dynamics by implementing a mixed/discontinuous Galerkin (MDG) finite element numerical scheme. We show that this methodology is stable, showing good accuracy and a remarkable scale invariance in its solution space. In addition the MDG method is robust, adapting well to various initial-boundary value problems of particular significance in a range of physical and chemical applications. We further show explicitly how to recover the Lagrangian frame (or pathline) solutions.

Discontinuous Galerkin methods for plasma physics in the scrape-off layer of tokamaks

“…These simulations were run with the same settings as above, but here we set p = 3 on 1024 finite elements. Frequently e-folding simulations are run on small domains with absorbing boundary conditions, but to avoid the potential first order boundary error that can arise in dispersive systems (see [43,53,54,74] for more details on this) we instead implement the same problem on a periodic domain (x, y) ∈ [0, 1600] × [0, 377], where a y-independent plasma source η(α 0 ) = 3α 0 e −(x−75) 2 /200 is added to the right hand side of (6.2). The resulting system is run until T = 50, 000 where saturation conditions are plainly evident.…”

“…In this regard we have implemented a fully discontinuous Galerkin method for solving (2.1)-(2.3). In contrast to mixed form finite element methods, for example, which use continuous Galerkin projections [53] to recover solutions to Poisson, our present approach preserves the discontinuous function spaces throughout the computation, thus expanding the well-posedness of the space of admissible solutions. Our formulation is novel, in that is supports multiples basis functions, and is run using modal, nodal, or mixed nodal/modal finite elements.…”

Modeling Chemical Reactors I: Quiescent Reactors

“…In these cases, the local propagation speed caused by the gradient of the concentration forces the Fick's component of the diffusion to obey a telegraph equation, which may often reduce to an integro-differential equation over all time [0, T ] coupled to a mass transport equation in the reactive components. These complications arise in systems that demonstrate large variations in concentrations over short time frames, though a large class of reactions demonstrate even more complicated and subtle behavior that might require the inclusion of quantum effects, such as in [48]. As a general rule we will not directly address these complications below, as we uniformly make the assumption that the reaction-diffusion system of equations employed is an appropriate approximate model for the system in question.…”

Adaptive hierarchic transformations for dynamically $p$-enriched slope-limiting over discontinuous Galerkin systems of generalized equations

“…For example, slope limiters are known to be of central importance in storm surge modeling [8,32] in order to obtain, for example, well-behaved solutions in the presence of complicated free-boundary conditions along adapting shorelines. Limiting regimes are also of substantial importance in quantum hydrodynamic systems [4,29] and surface wave models [26] where they are used to reduce the oscillations caused by mathematical dispersion terms (i.e. nonlinear third order spatial derivative terms) that pervade, for example, tunneling solutions.…”