Outflow Boundary Conditions for the Fourier Transformed One-Dimensional Vlasov–Poisson System

In order to facilitate numerical simulations of plasma phenomena where kinetic processes are important, we have studied the technique of Fourier transforming the Vlasov equation analytically in the velocity space, and solving the resulting equation numerically. Special attention has been paid to the boundary conditions of the Fourier transformed system. By using outgoing wave boundary conditions in the Fourier transformed space, small-scale information in velocity space is carried outside the computational domain and is lost. Thereby the so-called recurrence phenomenon is reduced. This method is an alternative to using numerical dissipation or smoothing operators in velocity space. Different high-order methods are used for computing derivatives as well as for the time-stepping, leading to an over-all fourth-order method.     

二维Helmholtz方程外问题基于自然边界归化的重叠型区域分解算法

１．引 言 设是平面光滑闭曲线,是以为边界的外部区域,考虑二维Ｈｅｌｍｈｏｌｔｚ方程外Ｎｅｕｍａｎｎ问题并在无穷远处满足Ｓｏｍｍｅｒｆｅｌｄ辐射条件其中是区域的边界的外法线方向,即指向由包围的内部区域．κ在许多情况下（例如约化波动方程）是实数,在另一些情况下则是纯虚数．本文仅讨论κ为纯虚数的情况,且不失一般性,可设Ｉｍ（ｋ）＞０． 用某些数值方法求解线性抛物型方程或线性双曲型方程的初边值问题时,可能导致求解Ｈｅｌｍｈｏｌｔｚ方程的外问题．例如,用自然边界元法求解线性抛物型方程的初边值问题时就导致求…     

Molecular dynamics simulations of oscillatory flows in microfluidic channels

In this paper we apply the direct non-equilibrium molecular dynamics technique to oscillatory flows of fluids in microscopic channels. Initially, we show that the microscopic simulations resemble the macroscopic predictions based on the Navier–Stokes equation very well for large channel width, high density and low temperature. Further simulations for high temperature and low density show that the non-slip boundary condition traditionally used in the macroscopic equation is greatly compromised when the fluid–wall interactions are the same as the fluid–fluid interactions. Simulations of a system with very narrow channel width confirm earlier findings of Poiseuille flow, namely, that the velocity profiles are modulated. We find that these modulations cannot be explained by the local area density model.

高性能容错处理核心的研究

本文提出了以三模冗余表决为基础构造容错处理核心的方案。三个ＣＰＵ模块采取松散同步的形式,只有在发生总线操作时才进行同步表决。在处理核心中采用了两级故障检测机制,以三模表决器作为第一级来保证处理核心有足够高的故障覆盖率;以ＣＰＵＡ模块的自检作为第二级来弥补表决器发现故障延迟的不足,缩短故障的潜伏时间。在表决器的设计中采取了表决器旁路及先编码、后表决的方法。先用编码的方式提取需要表决的数据的特征,然后     

Numerical implementation of the asymptotic boundary conditions for steadily propagating 2D solitons of Boussinesq type equations

In the present paper, a difference scheme on a non-uniform grid is constructed for the stationary propagating localized waves of the 2D Boussinesq equation in an infinite region. Using an argument stemming form a perturbation expansion for small wave phase speeds, the asymptotic decay of the wave profile is identified as second-order algebraic. For algebraically decaying solution a new kind of nonlocal boundary condition is derived, which allows to rigorously project the asymptotic boundary condition at the boundary of a finite-size computational box. The difference approximation of this condition together with the bifurcation condition complete the algorithm. Numerous numerical validations are performed and it is shown that the results comply with the second-order estimate for the truncation error even at the boundary lines of the grid. Results are obtained for different values of the so-called ‘rotational inertia’ and for different subcritical phase speeds. It is found that the limits of existence of the 2D solution roughly correspond to the similar limits on the phase speed that ensure the existence of subcritical 1D stationary propagating waves of the Boussinesq equation.     

A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ? by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.     

On the complexity of a two-point boundary value problem in different settings

We study the complexity of a two-point boundary value problem. We concentrate on the linear problem of order k with separated boundary conditions. Right-hand side functions are assumed to be r times differentiable with all derivatives bounded by a constant. We consider three models of computation: deterministic with standard and linear information, randomized and quantum. In each setting, we construct an algorithm for solving the problem, which allows us to establish upper complexity bounds. In the deterministic setting, we show that the use of linear information gives us a speed-up of at least one order of magnitude compared with the standard information. For randomized algorithms, we show that the speed-up over standard deterministic algorithms is by 1/2 in the exponent. For quantum algorithms, we can achieve a speed-up by one order of magnitude. We also provide lower complexity bounds. They match upper bounds in the deterministic setting with the standard information, and almost match upper bounds in the randomized and quantum settings. In the deterministic setting with the linear information, a gap still remains between the upper and lower complexity bounds.     

Richardson extrapolation applied to boundary element method results in a Dirichlet problem for the Laplace equation

Richardson extrapolation is used to improve the accuracy of the numerical solutions for the normal boundary flux and for the interior potential resulting from the boundary element method. The boundary integral equations arise from a direct boundary integral formulation for solving a Dirichlet problem for the Laplace equation. The Richardson extrapolation is used in two different applications: (i) to improve the accuracy of the collocation solution for the normal boundary flux and, separately, (ii) to improve the solution for the potential in the domain interior. The main innovative aspects of this work are that the orders of dominant error terms are estimated numerically, and that these estimates are then used to develop an a posteriori technique that predicts if the Richardson extrapolation results for applications (i) and (ii) are reliable. Numerical results from test problems are presented to demonstrate the technique.     

High-accuracy quadrature methods for solving boundary integral equations of steady-state anisotropic heat conduction problems with Dirichlet conditions

This paper presents the mechanical quadrature methods (MQMs) for solving the boundary integral equations of steady-state anisotropic heat conduction equation on the smooth domains and polygons, respectively. The costless and high-accurate Sidi–Israeli quadrature formula are applied to deal with the integrals in which the kernels have a logarithmic singularity. Especially, the Sidi transformation is used for the polygon cases in order to obtain a rapid convergence by degrading the singularity at the corners on the boundary. The convergence and stability of the MQMs solution are proved based on Anselone's collective compact theory. In addition, asymptotic error expansion of the MQMs shows that the approximation order is of O(h³), where h is the partition size of the boundary. Finally, numerical examples are tested and results verify the theoretical analysis.