Hybrid molecular-continuum methods for micro- and nanoscale liquid flows

This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.Many flows at microscale and below are characterised by an inherent multiscale nature and accurate numerical modelling of the phenomena involved is the cornerstone for enhancing the applicability of micro and nanofluidics in the industrial environment. This paper presents a hybrid molecular-continuum strategy named as point wise coupling for studying complex micro- and nanoscale flows. In this strategy one performs continuum simulations and uses a molecular solver for computing flow properties. The hybrid methodology utilises a numerical procedure to minimise the cost of the computationally expensive molecular solver. Simulations have been carried out for a slip Poiseuille flow test case. The hybrid results are in good agreement with analytical solutions and pervious molecular simulations.This study is funded by the EPSRC, MoD and AWE through the grant EP/D051940-JGS 607, as well as from the European Commission under the 6th Framework Program (Project: DINAMICS, NMP4-CT-2007-026804)

Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Some numerical methods for solving stochastic impulse control in natural gas storage facilities

The valuation of gas storage facilities is characterized as a stochastic impulse control problem with finite horizon resulting in Hamilton-Jacobi-Bellman (HJB) equations for the value function. In this context the two catagories of solving schemes for optimal switching are discussed in a stochastic control framework. We reviewed some numerical methods which include approaches related to partial differential equations (PDEs), Markov chain approximation, nonparametric regression, quantization method and some practitioners’ methods. This paper considers optimal switching problem arising in valuation of gas storage contracts for leasing the storage facilities, and investigates the recent developments as well as their advantages and disadvantages of each scheme based on dynamic programming principle (DPP

A general solution to the Schr\"odinger-Poission equation for charged hard wall: Application to potential profile of an AlN/GaN barrier structure

A general, system-independent formulation of the parabolic Schr\"odinger-Poisson equation is presented for a charged hard wall in the limit of complete screening by the ground state. It is solved numerically using iteration and asymptotic-boundary conditions. The solution gives a simple relation between the band bending and charge density at an interface. I further develop approximative analytical forms for the potential and wave function, based on properties of the exact solution. Specific tests of the validity of the assumptions leading to the general solution are made. The assumption of complete screening by the ground state is found be a limitation; however, the general solution still provides a fair approximate account of the potential when the bulk is doped. The general solution is further used in a simple model for the potential profile of an AlN/GaN barrier, and gives an approximation which compares well with the solution of the full Schr\"odinger-Poisson equation