Quantum Hydrodynamics with Trajectories: The Nonlinear Conservation Form Mixed/Discontinuous Galerkin Method with Applications in Chemistry

TL;DR

This paper introduces a stable and accurate mixed/discontinuous Galerkin finite element method for solving quantum hydrodynamic equations in chemical dynamics, capable of recovering Lagrangian solutions and applicable to diverse initial-boundary problems.

Contribution

It develops a novel MDG numerical scheme for quantum hydrodynamics that demonstrates stability, accuracy, and scale invariance, with explicit Lagrangian frame recovery.

Findings

Method is stable and accurate across various problems

Solution exhibits remarkable scale invariance

Lagrangian solutions can be explicitly recovered

Abstract

We 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.