A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ? _k ³ ?? _k?1 elements whereas the magnetic part of the equations is approximated by discontinuous ? _k ³ ?? _k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.     

A Dynamic Penalty or Projection Method for Incompressible Fluids

In a previous work (Angot et al. in J. Comput. Appl. Math. 226:228–245, 2009), some penalty–projection methods have been tested for the numerical analysis of the Navier-Stokes equations. The purpose of this study is to introduce a variant of the penalty–projection method which allows us to compute the solutions faster than by using the previous solver. This new variant combines dynamically and alternatively a penalty procedure and a projection procedure according to the size of the divergence of the velocity. In other words, this study aims to prove that it is possible to project the intermediate velocity, computed by the first step of the penalty–projection method, only if its divergence is larger than a specified threshold. Theoretical estimates for the new method are given, which are in accordance with the numerical results provided.     

三维Navier-Stokes方程分步法的并行算法在异构平台上实现初探

本文选取了三维不可压缩流动方程的分步法(fractional-step method),其中动量方程使用BiCGSTAB算法进行迭代求解,而压力泊松方程使用Fourier变换法进行直接求解。本文研究该算法在集群平台上的并行算法,从区域分解入手,分析一维、两维、三维区域划分三种情况下,各并行处理器上的计算量与通讯量,根据分析结果使用两维区域分解。分析BiCGSTAB算法和泊松Fourier变换法在GPGPU异构平台上的移植方法。最后,本文分析了BiCGSTAB和泊松方程Fourier变换法两种算法在CPU集群和GPGPU异构平台上的并行性能结果。     

An Equal-Order DG Method for the Incompressible Navier-Stokes Equations

We introduce and analyze a discontinuous Galerkin method for the incompressible Navier-Stokes equations that is based on finite element spaces of the same polynomial order for the approximation of the velocity and the pressure. Stability of this equal-order approach is ensured by a pressure stabilization term. A simple element-by-element post-processing procedure is used to provide globally divergence-free velocity approximations. For small data, we prove the existence and uniqueness of discrete solutions and carry out an error analysis of the method. A series of numerical results are presented that validate our theoretical findings.     

A Pseudospectral Multi-Domain Method for the Incompressible Navier–Stokes Equations

A pseudospectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method is designed for moderately complex geometries by means of a multi-domain approach. Key components are a Chebyshev collocation discretization, a special pressure-correction scheme and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the method with respect to the multi-domain functionality is investigated and compared to finite-volume approaches.     

A Parallel Subgrid Stabilized Finite Element Method Based on Two-Grid Discretization for Simulation of 2D/3D Steady Incompressible Flows

Based on domain decomposition and two-grid discretization, a parallel subgrid stabilized finite element method for simulation of 2D/3D steady convection dominated incompressible flows is proposed and analyzed. In this method, a subgrid stabilized nonlinear Navier–Stokes problem is first solved on a coarse grid where the stabilization term is based on an elliptic projection defined on the same coarse grid, and then corrections are calculated in overlapped fine grid subdomains by solving a linearized problem. By the technical tool of local a priori estimate for finite element solution, error bounds of the approximate solution are estimated. Algorithmic parameter scalings of the method are derived. Numerical results are also given to demonstrate the effectiveness of the method.     

A Compact Higher Order Finite Difference Method for the Incompressible Navier–Stokes Equations

A numerical method based on compact fourth order finite difference approximations is used for the solution of the incompressible Navier–Stokes equations. Our method is implemented for two dimensional, curvilinear coordinates on orthogonal, staggered grids. Two numerical experiments confirm the theoretically expected order of accuracy.     

A Semi-smooth Newton Method for Regularized State-constrained Optimal Control of the Navier-Stokes Equations

In this paper, we study semi-smooth Newton methods for the numerical solution of regularized pointwise state-constrained optimal control problems governed by the Navier-Stokes equations. After deriving an appropriate optimality system for the original problem, a class of Moreau-Yosida regularized problems is introduced and the convergence of their solutions to the original optimal one is proved. For each regularized problem a semi-smooth Newton method is applied and its local superlinear convergence verified. Finally, selected numerical results illustrate the behavior of the method and a comparison between the max-min and the Fischer-Burmeister as complementarity functionals is carried out.     

A Parallel Domain Decomposition Method for 3D Unsteady Incompressible Flows at High Reynolds Number

Numerical simulation of three-dimensional incompressible flows at high Reynolds number using the unsteady Navier–Stokes equations is challenging. In order to obtain accurate simulations, very fine meshes are necessary, and such simulations are increasingly important for modern engineering practices, such as understanding the flow behavior around high speed trains, which is the target application of this research. To avoid the time step size constraint imposed by the CFL number and the fine spacial mesh size, we investigate some fully implicit methods, and focus on how to solve the large nonlinear system of equations at each time step on large scale parallel computers. In most of the existing implicit Navier–Stokes solvers, segregated velocity and pressure treatment is employed. In this paper, we focus on the Newton–Krylov–Schwarz method for solving the monolithic nonlinear system arising from the fully coupled finite element discretization of the Navier–Stokes equations on unstructured meshes. In the subdomain, LU or point-block ILU is used as the local solver. We test the algorithm for some three-dimensional complex unsteady flows, including flows passing a high speed train, on a supercomputer with thousands of processors. Numerical experiments show that the algorithm has superlinear scalability with over three thousand processors for problems with tens of millions of unknowns.     

Algebraic Pressure Segregation Methods for the Incompressible Navier-Stokes Equations

This work is an overview of algebraic pressure segregation methods for the incompressible Navier-Stokes equations. These methods can be understood as an inexact LU block factorization of the original system matrix. We have considered a wide set of methods: algebraic pressure correction methods, algebraic velocity correction methods and the Yosida method. Higher order schemes, based on improved factorizations, are also introduced. We have also explained the relationship between these pressure segregation methods and some widely used preconditioners, and we have introduced predictor-corrector methods, one-loop algorithms where nonlinearity and iterations towards the monolithic system are coupled. The first author’s research was supported by the European Community through the Marie Curie contract NanoSim (MOIF-CT-2006-039522).     

An Iterative Method for Solving Nonlinear Operator Equations Using Generalized Divided Differences

An iterative method that uses generalized divided differences to solve nonlinear operator equations is proposed. Local and semi local convergence of the proposed method is shown. Numerical examples are also presented to demonstrate the efficiency of the method.     

Interior Penalty Discontinuous Galerkin Method for Maxwell’s Equations in Cold Plasma

In this paper, we develop an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell’s equations in cold plasma. Both semi and fully discrete DG schemes are constructed, and optimal error estimates in the energy norm are proved. To our best knowledge, this is the first error analysis carried out for the DG method for Maxwell’s equations in dispersive media.     

3D and 2D/3D holograms model

The estimating problem of 3D holograms orientation selectivity on angular, orthogonal, and azimuthal sensitivity parameters is formulated and solved. Tenfold increase of density 3D, 2D/3D holograms in comparison with 2D holograms at given selectivity for ones is shown in theory and experimentally.     

一种具有RST不变性的2D/3D轨迹相似性度量

具有旋转、缩放、平移不变性的轨迹相似性度量是实现精准手语识别、相似轨迹检索等的关键环节,常规的相似性度量往往不满足这一要求,特别是不具备旋转不变性。提出一种具有旋转、缩放、平移不变性的轨迹相似性度量方法,该方法首先对轨迹进行滤波、归一化、等间距重采样等预处理操作,然后对任意两条待比较的轨迹估计最优旋转矩阵,从而消除旋转对距离度量的干扰。该方法对二维、三维轨迹数据均适用,计算复杂度为[ON],与曲率、挠率等不变量相比,该方法对轨迹噪声不敏感。     

A Finite Difference Method and Analysis for 2D Nonlinear Poisson–Boltzmann Equations

A fast finite difference method based on the monotone iterative method and the fast Poisson solver on irregular domains for a 2D nonlinear Poisson–Boltzmann equation is proposed and analyzed in this paper. Each iteration of the monotone method involves the solution of a linear equation in an exterior domain with an arbitrary interior boundary. A fast immersed interface method for generalized Helmholtz equations on exterior irregular domains is used to solve the linear equation. The monotone iterative method leads to a sequence which converges monotonically from either above or below to a unique solution of the problem. This monotone convergence guarantees the existence and uniqueness of a solution as well as the convergence of the finite difference solution to the continuous solution. A comparison of the numerical results against the exact solution in an example indicates that our method is second order accurate. We also compare our results with available data in the literature to validate the numerical method. Our method is efficient in terms of accuracy, speed, and flexibility in dealing with the geometry of the domain     

线性矩阵方程异类约束最小二乘解的迭代算法

多矩阵变量线性矩阵方程(LME)约束解的计算问题在参数识别、结构设计、振动理论、自动控制理论等领域都有广泛应用。本文借鉴求线性矩阵方程(LME)同类约束最小二乘解的迭代算法,通过构造等价的线性矩阵方程组,建立了求多矩阵变量LME的一种异类约束最小二乘解的迭代算法,并证明了该算法的收敛性。在不考虑舍入误差的情况下,利用该算法不仅可在有限步计算后得到LME的一组异类约束最小二乘解,而且选取特殊初始矩阵时,可求得LME的极小范数异类约束最小二乘解。另外,还可求得指定矩阵在该LME的异类约束最小二乘解集合中的最佳逼近解。算例表明,该算法是有效的。