Xiantao Li

Numerical Approximations of Pressureless and Isothermal Gas Dynamics

We study several schemes of first- or second-order accuracy based on kinetic approximations to solve pressureless and isothermal gas dynamics equations. The pressureless gas system is weakly hyperbolic, giving rise to the formation of density concentrations known as delta-shocks. For the isothermal gas system, the infinite speed of expansion into vacuum leads to zero timestep in the Godunov method based on exact Riemann solver. The schemes we consider are able to bypass these difficulties. They are proved to satisfy positiveness of density and discrete entropy inequalities, to capture the delta-shocks, and to treat data with vacuum.

TWO MOMENT SYSTEMS FOR COMPUTING MULTIPHASE SEMICLASSICAL LIMITS OF THE SCHRÖDINGER EQUATION

Two systems of hyperbolic equations, arising in the multiphase semiclassical limit of the linear Schrodinger equations, are investigated. One stems from a Wigner measure analysis and uses a closure by the Delta functions, whereas the other relies on the classical WKB expansion and uses the Heaviside functions for closure. The two resulting moment systems are weakly and non-strictly hyperbolic respectively. They provide two different Eulerian methods able to reproduce superimposed signals with a finite number of phases. Analytical properties of these moment systems are investigated and compared. Efficient numerical discretizations and test-cases with increasing difficulty are presented.

Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism

The Mori-Zwanzig formalism for coarse-graining a complex dynamical system typically introduces memory effects. The Markovian assumption of delta-correlated fluctuating forces is often employed to simplify the formulation of coarse-grained (CG) models and numerical implementations. However, when the time scales of a system are not clearly separated, the memory effects become strong and the Markovian assumption becomes inaccurate. To this end, we incorporate memory effects into CG modeling by preserving non-Markovian interactions between CG variables, and the memory kernel is evaluated directly from microscopic dynamics. For a specific example, molecular dynamics (MD) simulations of star polymer melts are performed while the corresponding CG system is defined by grouping many bonded atoms into single clusters. Then, the effective interactions between CG clusters as well as the memory kernel are obtained from the MD simulations. The constructed CG force field with a memory kernel leads to a non-Markovian dissipative particle dynamics (NM-DPD). Quantitative comparisons between the CG models with Markovian and non-Markovian approximations indicate that including the memory effects using NM-DPD yields similar results as the Markovian-based DPD if the system has clear time scale separation. However, for systems with small separation of time scales, NM-DPD can reproduce correct short-time properties that are related to how the system responds to high-frequency disturbances, which cannot be captured by the Markovian-based DPD model.

A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks

We present a multiscale model for numerical simulations of dynamics of crystalline solids. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling - the heterogeneous multiscale method (HMM). We derive an explicit coupling condition at the atomistic/continuum interface. Application to the dynamics of brittle cracks under various loading conditions is presented as test examples.

Data-driven parameterization of the generalized Langevin equation

Significance The generalized Langevin equation (GLE) provides a precise description of coarse-grained variable dynamics in reduced dimension models. However, computation of the memory kernel poses a major challenge to the practical use of the GLE. This paper presents a data-driven approach to compute the memory kernel, relying on a hierarchy of parameterized rational approximations in terms of the Laplace transform, which can be expanded to arbitrarily high order as necessary. This parameterization makes it convenient to represent the GLE via an extended stochastic model where the memory term is eliminated by properly introducing auxiliary variables. The present method is well-suited for constructing reduced models for nonequilibrium properties of complex systems such as biomolecules, chemical reaction networks, and climate simulations. We present a data-driven approach to determine the memory kernel and random noise in generalized Langevin equations. To facilitate practical implementations, we parameterize the kernel function in the Laplace domain by a rational function, with coefficients directly linked to the equilibrium statistics of the coarse-grain variables. We show that such an approximation can be constructed to arbitrarily high order and the resulting generalized Langevin dynamics can be embedded in an extended stochastic model without explicit memory. We demonstrate how to introduce the stochastic noise so that the second fluctuation-dissipation theorem is exactly satisfied. Results from several numerical tests are presented to demonstrate the effectiveness of the proposed method.

Optoelectronic control of spin dynamics at near-terahertz frequencies in magnetically doped quantum wells

We use time-resolved Kerr rotation to demonstrate the optical and electronic tuning of both the electronic and local moment (Mn) spin dynamics in electrically gated parabolic quantum wells derived from II-VI diluted magnetic semiconductors. By changing either the electrical bias or the laser energy, the electron spin precession frequency is varied from 0.1 to 0.8 THz at a magnetic field of 3 T and at a temperature of 5 K. The corresponding range of the electrically-tuned effective electron g-factor is an order of magnitude larger compared with similar nonmagnetic III-V parabolic quantum wells. Additionally, we demonstrate that such structures allow electrical modulation of local moment dynamics in the solid state, which is manifested as changes in the amplitude and lifetime of the Mn spin precession signal under electrical bias. The large variation of electron and Mn-ion spin dynamics is explained by changes in magnitude of the sp−d exchange overlap.

A generalized Irving–Kirkwood formula for the calculation of stress in molecular dynamics models

In non-equilibrium molecular dynamics simulations, continuum mechanics quantities can be computed from the position and momentum of the particles based on the classical Irving-Kirkwood formalism. For practical purposes, the implementations of Irving-Kirkwood formulas often involve a spatial averaging using a smooth kernel function. The resulting formula for the stress has been known as Hardy stress. Usually results obtained this way still need to be further processed to reduce the fluctuation, e.g., by ensemble or time averaging. In this paper we extend Hardys formulas by systematically incorporating both spatial and temporal averaging into the expression of continuum quantities. The derivation follows the Irving-Kirkwood formalism, and the average quantities still satisfy conservation laws in continuum mechanics. We will discuss the selection of kernel functions and present several numerical tests.

The Discontinuous Galerkin Method for the Multiscale Modeling of Dynamics of Crystalline Solids

We present a multiscale model for numerical simulation of dynamics of crystalline solids. The method couples nonlinear elastodynamics as the continuum description and molecular dynamics as another component at the atomic scale. The governing equations on the macroscale are solved by the discontinuous Galerkin method, which is built up with an appropriate local curl-free space to produce a coherent displacement field. The constitutive data are based on the underlying atomistic model: it is either calibrated prior to the computation or obtained from molecular dynamics as the computation proceeds. The decision to use either the former or the latter is made locally for each cell based on suitable criteria.

Variational boundary conditions for molecular dynamics simulations: Treatment of the loading condition

This paper aims to extend the variational boundary conditions for molecular dynamics simulation [X. Li, W. E, Variational boundary conditions for molecular dynamics simulations of solids at low temperature, Commun. Comp. Phys. 1 (2006) 136-176; X. Li, W. E, Boundary conditions for molecular dynamics simulations at finite temperature: treatment of the heat bath, Phys. Rev. B 76 (2007) 104107], to take into account external loading conditions. Two derivations of the exact boundary conditions are presented, one with Mori-Zwanzig projection procedure, and the other using lattice Greens functions. Approximate boundary conditions, which are more efficient in practice, are then discussed. Finally several numerical experiments are presented to demonstrate the effectiveness of these methods.

On the stability of boundary conditions for molecular dynamics

We study the stability of boundary conditions for molecular dynamics simulations. A general stability criterion is established. We first consider a one-dimensional model with nearest neighbor interaction and multiple-neighbor interactions. We then generalize the results to more realistic models in 3D with nonlinear atomic interaction.