Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form

Finite difference operators satisfying the summation-by-parts (SBP) rules can be used to obtain high order accurate, energy stable schemes for time-dependent partial differential equations, when the boundary conditions are imposed weakly by the simultaneous approximation term (SAT).In general, an SBP-SAT discretization is accurate of order p + 1 with an internal accuracy of 2p and a boundary accuracy of p. Despite this, it is shown in this paper that any linear functional computed from the time-dependent solution, will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way.The method does not involve the solution of the dual equations, and superconvergent functionals are obtained at no extra computational cost. Four representative model problems are analyzed in terms of convergence and errors, and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs.     

Error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations

This article considers a posteriori error estimation and anisotropic mesh refinement for three-dimensional laminar aerodynamic flow simulations. The optimal order symmetric interior penalty discontinuous Galerkin discretization which has previously been developed for the compressible Navier–Stokes equations in two dimensions is extended to three dimensions. Symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model. Using duality arguments, an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients. Furthermore, residual-based indicators as well as adjoint-based indicators for goal-oriented refinement are derived. These refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous Galerkin (DG) discretization. Two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint-based error estimation are presented. The performance of the proposed discretization, error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows.     

Memory in time-dependent density functional theory

Exact time-dependent density functionals remember both the entire history of the density and the initial wave function. We show that the two effects are intimately related, and all history dependence can be written as initial-state dependence, including that of the exchange-correlation kernel. For states that can be evolved from a ground state, all initial-state dependence is a dependence on a pseudo-prehistory, providing a route to excited-state densities from time-dependent density functional theory.     

First passage time problems in time-dependent fields

This paper discusses the simplest first passage time problems for random walks and diffusion processes on a line segment. When a diffusing particle moves in a time-varying field, use of the adjoint equation does not lead to any simplification in the calculation of moments of the first passage time as is the case for diffusion in a time-invariant field. We show that for a discrete random walk in the presence of a sinusoidally varying field there is a resonant frequency * for which the mean residence time on the line segment is a minimum. It is shown that for a random walk on a line segment of lengthL the mean residence time goes likeL ² for largeL when *, but when =* the dependence is proportional toL. The results of our simulation are numerical, but can be regarded as exact. Qualitatively similar results are shown to hold for diffusion processes by a perturbation expansion in powers of a dimensionless velocity. These results are extended to higher values of this parameter by a numerical solution of the forward equation.     

非定常辐射输运问题的蒙特卡罗自适应偏倚抽样

用蒙特卡罗(MC)方法模拟高温、高压、多介质、大变形辐射输运问题时,由于网格体积悬殊,导致各网格通量的统计误差涨落很大,随着时间步的增加,误差积累甚至会导致计算结果失真.为此,发展了针对全局网格计算的源偏倚抽样技巧.用于源偏倚抽样的价值函数基于上个时间步各网格通量及误差,通过加权构造产生,它比传统MC通过解伴随方程获取价值的性价比要高得多.数值试验表明,全局源偏倚抽样通过自适应分配当前时间步各网格的粒子数,有效地降低了当前步重要网格通量误差. 关键词： 非定常 辐射输运 蒙特卡罗 源偏倚抽样     

Error estimates and specification parameters for functional renormalization

We present a strategy for estimating the error of truncated functional flow equations. While the basic functional renormalization group equation is exact, approximated solutions by means of truncations do not only depend on the choice of the retained information, but also on the precise definition of the truncation. Therefore, results depend on specification parameters that can be used to quantify the error of a given truncation. We demonstrate this for the BCS–BEC crossover in ultracold atoms. Within a simple truncation the precise definition of the frequency dependence of the truncated propagator affects the results, indicating a shortcoming of the choice of a frequency independent cutoff function.     

Excited states dynamics in time-dependent density functional theory

We present numerical simulations of femtosecond laser induced dynamics of some selected simple molecules -- hydrogen, singly ionized sodium dimer, singly ionized helium trimer and lithium cyanide. The simulations were performed within a real-space, real-time, implementation of time-dependent density functional theory (TDDFT). High harmonic generation, Coulomb explosion and laser induced photo-dissociation are observed. The scheme also describes non-adiabatic effects, such as the appearance of even harmonics for homopolar but isotopically asymmetric dimers, even if the ions are treated classically. This TDDFT-based method is reliable, scalable, and extensible to other phenomena such as photoisomerization, molecular transport and chemical reactivity.Received: 15 October 2003PACS: 33.80.Gj Diffuse spectra; predissociation, photodissociation - 33.80.Wz Other multiphoton processes     

A non-self-consistent range-separated time-dependent density functional approach for large-scale simulations

We propose an efficient method for carrying out time-dependent density functional theory (TDDFT) calculations using range-separated hybrid exchange-correlation functionals. Based on a non-self-consistent range-separated Hamiltonian, the method affords large-scale simulations at a fraction of the computational time of conventional hybrid TDDFT approaches. For typical benchmark molecules including N(2), CO, C(6)H(6), H(2)CO and the C(2)H(4)-C(2)F(4) dimer, the method possesses the same level of accuracy as the conventional approaches for the valence, Rydberg, and charge-transfer excitation energies when compared to the experimental results. The method is used to determine π → π* excitations in both disordered and crystalline poly(3-hexylthiophene) (P3HT) conjugated polymers with more than six hundred atoms and it yields excitation energies and charge densities that are in excellent agreement with experiments. The simulation of the crystalline P3HT reveals that the phase of the wavefunctions could have an important effect on the excitation energy; a hypothesis based on π-π stacking is proposed to explain this novel effect in conjugated polymers.     

一般含时线性势的量子解及有关问题

对于一般形式的含时线性势, 通过假设波函数形式的方法得到了Schr?dinger方程的精确和完备解. 同时指出, 用两个波函数φ(t)〉和ψ(t)〉定义的坐标和动量的矩阵元〈φ(t)xψ(t)〉和〈φ(t)pψ(t)〉满足经典形式的运动方程. 按照量子力学的系综理论, 这样的经典形式的运动方程实际上是流体方程. 进一步研究发现, 对于任意形式的线性系统有类似的结论. 关键词： 线性势 Schr?dinger方程 Heisenberg对应原理 经典运动方程     

Kinetic model analysis of time-dependent problems in polyatomic gases

In this work we analyze time-dependent problems like sound propagation and light scattering in dilute polyatomic gases by using a kinetic model of the Boltzmann equation that replaces the collision operator by a single relaxation-time term which is compatible with Grad's 6-moment approximation. Comparison of the theoretical results with available experimental data in nitrogen, oxygen, carbon dioxide and methane shows that the model equation can be used to describe the acoustic properties and the light scattering spectrum of polyatomic gases in both hydrodynamic and kinetic regimes as long as the external oscillation frequency is smaller than the frequency required for the translational and the internal degrees of freedom to come to thermal equilibrium.     

Speckle tracking for multi-dimensional flow estimation

Speckle tracking methods overcome the major limitations of current Doppler methods for flow imaging and quantification: angle dependence and aliasing. In this paper, we review the development of speckle tracking, with particular attention to the advantages and limitations of two-dimensional algorithms that use a single transducer aperture. Ensemble tracking, a recent speckle tracking method based upon parallel receive processing, is described. Experimental results with ensemble tracking indicate the ability to measure laminar flow in a phantom at a beam-vessel angle of 60 degrees, which had not been possible with previous 2D speckle tracking methods. Finally, important areas for future research in speckle tracking are briefly summarized.     

Elastic capsules in shear flow: Analytical solutions for constant and time-dependent shear rates

We investigate the dynamics of microcapsules in linear shear flow within a reduced model with two degrees of freedom. In previous work for steady shear flow, the dynamic phases of this model, i.e. swinging, tumbling and intermittent behaviour, have been identified using numerical methods. In this paper, we integrate the equations of motion in the quasi-spherical limit analytically for time-constant and time-dependent shear flow using matched asymptotic expansions. Using this method, we find analytical expressions for the mean tumbling rate in general time-dependent shear flow. The capsule dynamics is studied in more detail when the inverse shear rate is harmonically modulated around a constant mean value for which a dynamic phase diagram is constructed. By a judicious choice of both modulation frequency and phase, tumbling motion can be induced even if the mean shear rate corresponds to the swinging regime. We derive expressions for the amplitude and width of the resonance peaks as a function of the modulation frequency.     

Vesicle dynamics in time-dependent elongation flow: wrinkling instability

We present experimental results on the relaxation dynamics of vesicles subjected to a time-dependent elongation flow. We observed and characterized a new instability, which results in the formation of higher-order modes of the vesicle shape (wrinkles), after a switch in the direction of the velocity gradient. This surprising generation of membrane wrinkles can be explained by the appearance of a negative surface tension during the vesicle deflation, which tunes itself to alternating stress. Moreover, the formation of buds in the vesicle membrane was observed in the vicinity of the dynamical transition point.     

First-principles simulations for attosecond photoelectron spectroscopy based on time-dependent density functional theory

We develop a first-principles simulation method for attosecond time-resolved photoelectron spectroscopy. This method enables us to directly simulate the whole experimental processes, including excitation, emission and detection on equal footing. To examine the performance of the method, we use it to compute the reconstruction of attosecond beating by interference of two-photon transitions (RABBITT) experiments of gas-phase Argon. The computed RABBITT photoionization delay is in very good agreement with recent experimental results from [Klünder et al., Phys. Rev. Lett. 106, 143002 (2011)] and [Guénot et al., Phys. Rev. A 85, 053424 (2012)]. This indicates the significance of a fully-consistent theoretical treatment of the whole measurement process to properly describe experimental observables in attosecond photoelectron spectroscopy. The present framework opens the path to unravel the microscopic processes underlying RABBITT spectra in more complex materials and nanostructures.     

Computational studies of CO and CO: Density functional theory and time-dependent density functional theory

Potential energy curves, equilibrium interatomic distances, term energies and harmonic vibration frequencies for the 16 lowest states of neutral carbon monoxide and the six lowest states of singly ionized carbon monoxide are calculated by density functional theory (DFT) and linear-response time-dependent density functional (LR-TDDFT) theory. The results are compared with experimental data. The two theories, DFT and LR-TDDFT, are described briefly.     

势问题的无单元Galerkin方法的误差估计

在高维情况下,首先研究了无单元Galerkin方法的形函数构造方法——移动最小二乘法在Sobolev空间W^k,p(Ω)中的误差估计.然后,在势问题的无单元Galerkin方法的基础上,研究了势问题的通过罚函数法施加本质边界条件的无单元Galerkin方法在Sobolev空间中的误差估计.当节点和形函数满足一定条件时,证明了该误差估计是最优阶的.从误差分析中可以看出,数值解的误差与权函数的影响半径密切相关.最后,通过算例验证了结论的正确性. 关键词： 无网格方法 无单元Galerkin方法 势问题 误差估计     

Equations of the surface harmonics method for solving time-dependent neutron transport problems and their verification

Time-dependent equations of the surface harmonics method (SHM) are obtained for planar one-dimensional geometry. The equations are verified by calculations of test problems from Benchmark Problem Book ANL-7416, and the capabilities and efficiency of applying the SHM for solving the time-dependent neutron transport equation in the diffusion approximation are demonstrated. The results of the work show that the implementation of the SHG for full-scale computations will make possible substantial progress in the efficient solution of time-dependent problems of neutron transport in nuclear reactors.