The boundary plasma turbulence code BOUT models tokamak boundary-plasma turbulence in a realistic divertor geometry using modified Braginskii equations for plasma vorticity, density (n_i), electron and ion temperature (T_e, T_i) and parallel momenta. The BOUT code solves for the plasma fluid equations in a three dimensional (3D) toroidal segment (or a toroidal wedge), including the region somewhat inside the separatrix and extending into the scrape-off layer; the private flux region is also included. In this paper, a description is given of the sophisticated physical models, innovative numerical algorithms, and modern software design used to simulate edge plasmas in magnetic fusion energy devices. The BOUT code's unique capabilities and functionality are exemplified via simulations of the impact of plasma density on tokamak edge turbulence and blob dynamics.

The derivation of the quantum lattice Boltzmann model is reviewed with special emphasis on recent developments of the model, namely, the extension to a multi-dimensional formulation and the application to the computation of the ground state of the Gross-Pitaevskii equation (GPE). Numerical results for the linear and nonlinear Schrödinger equation and for the ground state solution of the GPE are also presented and validated against analytical results or other classical schemes such as Crank-Nicholson. 

Based on the high order essentially non-oscillatory (ENO) Lagrangian type scheme on quadrilateral meshes presented in our earlier work [3], in this paper we develop a third order conservative Lagrangian type scheme on curvilinear meshes for solving the Euler equations of compressible gas dynamics. The main purpose of this work is to demonstrate our claim in [3] that the accuracy degeneracy phenomenon observed for the high order Lagrangian type scheme is due to the error from the quadrilateral mesh with straight-line edges, which restricts the accuracy of the resulting scheme to at most second order. The accuracy test given in this paper shows that the third order Lagrangian type scheme can actually obtain uniformly third order accuracy even on distorted meshes by using curvilinear meshes. Numerical examples are also presented to verify the performance of the third order scheme on curvilinear meshes in terms of resolution for discontinuities and non-oscillatory properties.

The main obstacle in sequential multiscale modeling is the pre-computation of the constitutive relation which often involves many independent variables. The constitutive relation of a polymeric fluid is a function of six variables, even after making the simplifying assumption that stress depends only on the rate of strain. Precomputing such a function is usually considered too expensive. Consequently the value of sequential multiscale modeling is often limited to "parameter passing". Here we demonstrate that sparse representations can be used to drastically reduce the computational cost for precomputing functions of many variables. This strategy dramatically increases the efficiency of sequential multiscale modeling, making it very competitive in many situations.

We investigate the validity of stationary simulations for semiconductor quantum charge transport in a one-dimensional resonant tunneling diode via fluid type models. Careful numerical investigations to a quantum hydrodynamic model reveal that the transient simulations do not always converge to the steady states. In particular, growing oscillations are observed at relatively large applied voltage. A dynamical bifurcation is responsible for the stability interchange of the steady state. Transient and stationary computations are also performed for a unipolar quantum drift-diffusion model.

Consider a time-harmonic electromagnetic plane wave incident on a microscopic semiconductor. Inside the medium, at any given frequency ω, more than one polariton mode can arise with the same frequency but different wavenumbers due to the presence of excitons. Besides Maxwell's boundary conditions, additional boundary conditions are required to handle the multi-mode polariton. In order to model the confinement effect of excitons in the microscopic semiconductor, Maxwell's equations and the Schrödinger equation are coupled to characterize the polarization in terms of the quantum description. In the weak confinement regime, we derive a perturbed dispersive dielectric constant by taking the exciton effect into account. We also analyze and compute the optical linear response of the exciton in both one-dimensional and two-dimensional confinements. For the one-dimensional case, the existence and uniqueness of the analytical solution are established in the resonance region. A finite difference method is developed to compute the two dimensional confinement. 

In an early approach, we proposed a kinetic model with multiple translational temperature [K. Xu, H. Liu and J. Jiang, Phys. Fluids 19, 016101 (2007)]. Based on this model, the stress strain relationship in the Navier-Stokes (NS) equations is replaced by the translational temperature relaxation terms. The kinetic model has been successfully used in both continuum and near continuum flow computations. In this paper, we will further validate the multiple translational temperature kinetic model to flow problems in multiple dimensions. First, a generalized boundary condition incorporating the physics of Knudsen layer will be introduced into the model. Second, the direct particle collision with the wall will be considered as well for the further modification of particle collision time, subsequently a new effective viscosity coefficient will be defined. In order to apply the kinetic model to near continuum flow computations, the gas-kinetic scheme will be constructed. The first example is the pressure-driven Poiseuille flow at Knudsen number 0.1, where the anomalous phenomena between the results of the NS equations and the Direct Simulation Monte Carlo (DSMC) method will be resolved through the multiple temperature model. The so-called Burnett-order effects can be captured as well by algebraic temperature relaxation terms. Another test case is the force-driven Poiseuille flow at various Knudsen numbers. With the effective viscosity approach and the generalized second-order slip boundary condition, the Knudsen minimum can be accurately obtained. The current study indicates that it is useful to use multiple temperature concept to model the non-equilibrium state in near continuum flow limit. In the continuum flow regime, the multiple temperature model will automatically recover the single temperature NS equations due to the efficient energy exchange in different directions.

The finite element method is a promising method for electronic structure calculations. In this paper, a new parallel mesh refinement method for electronic structure calculations is presented. Some properties of the method are investigated to make it more efficient and more convenient for implementation. Several practical issues such as distributed memory parallel computation, less tetrahedra prototypes, and the assignment of the mesh elements carried out independently in each sub-domain will be discussed. The numerical experiments on the periodic system, cluster and nano-tube are presented to demonstrate the effectiveness of the proposed method.

We construct a numerical scheme based on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction (GTD) to simulate the high frequency linear waves diffracted by a half plane. We first introduce a condition, based on the GTD theory, at the vertex of the half plane to account for the diffractions, and then build in this condition as well as the reflection boundary condition into the numerical flux of the geometrical optics Liouville equation. Numerical experiments are used to verify the validity and accuracy of this new Eulerian numerical method which is able to capture the moments of high frequency and diffracted waves without fully resolving the high frequency numerically. 

This paper is concerned with the pattern dynamics of the generalized nonlinear Schrödinger equations (NSEs) related with various nonlinear physical problems in plasmas. Our theoretical and numerical results show that the higher-order nonlinear effects, acting as a Hamiltonian perturbation, break down the NSE integrability and lead to chaotic behaviors. Correspondingly, coherent structures are destroyed and replaced by complex patterns. Homoclinic orbit crossings in the phase space and stochastic partition of energy in Fourier modes show typical characteristics of the stochastic motion. Our investigations show that nonlinear phenomena, such as wave turbulence and laser filamentation, are associated with the homoclinic chaos. In particular, we found that the unstable manifolds W^(u) possessing the hyperbolic fixed point correspond to an initial phase θ =45^◦ and 225^◦, and the stable manifolds W^(s) correspond to θ=135^◦ and 315^◦. 

Ultra-parallel flow simulations on hundreds of thousands of processors require new multi-level domain decomposition methods. Here we present such a new two-level method that has features both of discontinuous and continuous Galerkin formulations. Specifically, at the coarse level the domain is subdivided into several big patches and within each patch a spectral element discretization (fine level) is employed. New interface conditions for the Navier-Stokes equations are developed to connect the patches, relaxing the C⁰continuity and minimizing data transfer at the patch interface. We perform several 3D flow simulations of a benchmark problem and of arterial flows to evaluate the performance of the new method and investigate its accuracy.

Enabling technologies are those technologies preparing input data, analyzing output data and facilitating the whole processes for numerical simulations. This paper outlines current enabling technologies for large-scale multidisciplinary simulations used in the High End Digital Prototyping (HEDP) system, a problem solving environment equipped with capability of mesh generation and large-scale visualization. A problem solving environment is a computer system that provides all the computational facilities necessary to solve a target class of problems. Mesh generation continues to be the pacing technology for a practical numerical analysis, which is essential to yielding an accurate and efficient solution. Large-scale visualization maps the massive data to some kinds of scenes interactively, which can be realized through a tiled display wall system with distributed visualization capability. HEDP is designed for large-scale and multidisciplinary simulations, and there are four categories of modules involved, namely pre-processing module, computing module, post-processing module, and platform control module. All these modules are coupled through a software bus, which makes the modules integrated seamlessly. Detailed design principles and applications of the HEDP environment are addressed in this paper.

We consider the extended Doi model for nematic liquid crystalline polymers in-planar shear flow, which is inhomogeneous in shear direction. We study the formation of microstructure and the dynamics of defects. We discretize the Fokker-Plank equation using the spherical harmonic spectral method. Five in-plane flow modes and eight out-of-plane flow modes are replicated in our simulations. In order to demonstrate the validity of our method in simulating liquid crystal dynamics, we replicated weak shear limit results and detected defects. We also demonstrate numerically that the Bingham closure model, which maintains energy dissipation, is a reliable closure model.

Based on the theory of optimization, we use edges and angles of cells to represent the geometric quality of computational grids, employ the local gradients of the flow variables to describe the variation of flow field, and construct a multi-objective programming model. The solution of this optimization problem gives appropriate balance between the geometric quality and adaptation of grids. By solving the optimization problem, we propose a new grid rezoning method, which not only keeps good geometric quality of grids, but also can track rapid changes in the flow field. In particular, it performs well for some complex concave domains with corners. We also incorporate the rezoning method into an Arbitrary Lagrangian-Eulerian (ALE) method which is widely used in the simulation of high-speed multi-material flows. The proposed rezoning and ALE methods of this paper are tested by a number of numerical examples with complex concave domains and compared with some other rezoning methods. The numerical results validate the robustness of the proposed methods.

An improved three-field gyrofluid model is proposed to numerically simulate ion-scale turbulence in tokamak plasmas, which includes the nonlinear evolution of perturbed electrostatic potential, parallel ion velocity and ion pressure with adiabatic electron response. It is benchmarked through advancing a gyrofluid toroidal global (GFT_G) code as well as the local version (GFT_L), with the emphasis of the collisionless damping of zonal flows. The nonlinear equations are solved by using Fourier decomposition in poloidal and toroidal directions and semi-implicit finite difference method along radial direction. The numerical implementation is briefly explained, especially on the periodic boundary condition in GFT_L version. As a numerical test and also practical application, the nonlinear excitation of geodesic acoustic mode (GAM), as well as its radial structure, is investigated in tokamak plasma turbulence.

With one- and two-dimensional particle-in-cell (PIC) codes, we simulate the generation of high power terahertz (THz) emission from the interaction of ultrashort intense lasers with tenuous plasma and gas targets. By driving high-amplitude electron plasma waves either with a laser wakefield or the beatwave of two laser pulses, powerful THz electromagnetic pulses can be produced by linear mode conversion in inhomogeneous plasma or by the transient current induced at the surfaces of a thin plasma layer of few plasma wavelengths. Even with incident lasers at moderate intensity such as 10¹⁷W/cm², the produced emission can be at the level of tens of MW in power and capable of affording field strengths of a few MV/cm, suitable for the studies of THz nonlinear physics. With field ionization included in the PIC codes, THz emission from laser interaction with tenuous gas targets is simulated. It is found that the transient transverse current formed during the ionization processes is responsible for the THz emission. With this mechanism, one may also obtain THz fields of MV/cm at lower laser intensity as compared with the schemes of plasma-wave excitation.

A multi-timescale algorithm is proposed for simulating time-dependent problems in micro- and nano- fluidics. The total simulation domain is spatially decomposed into two regions. Molecular dynamics is employed in the crucial interfacial regions and continuum hydrodynamics is adopted in the remaining bulk regions. The coupling is through “constrained dynamics” in an overlap region. Our time scheme is based on the time scale separation between the continuum macro time step and molecular micro time step. This allows the molecular dynamics during one macro time step to be treated as in quasi-steady state. Therefore, molecular simulation is only performed in two shorter time intervals. Through linear extrapolation of macroscopic velocities and re-initialization of particle configurations, we can significantly reduce the total computational cost. We demonstrate and discuss our time algorithm through hybrid simulation of channel flow driven by a sinusoidally moving top wall. Converging results are achieved for cases of large separation of time scale with much less computational cost than with the original hybrid simulation without time extrapolation.