对流-扩散问题的特征──块中心差分法

１．引言１９８２年,Ｄｏｕｇｌａｓ和Ｒｕｓｓｅｌｌ［１］提出解对流一扩散问题的特征一差分方法,网格节点为均匀分布,求解区域为直线Ｒ．文中讨论了基于二次插值的特征一差分格式,但其近似解按离散Ｌ２模未达到最优阶误差估计．１９８８年Ｗｅｉｓｅｒ和Ｗｈｅｅｌｅｒ［２］提出解线性椭圆型和线性抛物型方程的块中心差分法,１９９１年王申林［３］讨论了解拟线性双曲型积分微分方程的块中心差分方法,其共同特点为近似解按离散的Ｌ２模达到最优阶误差估计,解的一阶导数的近似解达到超收敛误差估计．１９９３年由同顺［４］讨论了…     

竞争法免疫层析试条的数学模型与仿真研究

根据竞争法免疫层析试条的检测原理,本研究将其分为TwA和TnA两种模式,结合对流扩散方程分别建立其数学模型,并运用COMSOL软件对试条的动态反应过程进行仿真,得到检测线和质控线上各复合物浓度关于各影响因素的关系曲线.在TwA模式中,分析了待测物浓度[A0]=0~20 mol/L,标记物浓度[P0]=0.01~100 mol/L,检测线位于5~20 mm位置等因素对于检测结果的影响;在TnA模式中,分析了[A0]=0~20 mol/L, [P0]=0.01~100 mol/L, [A0]和[P0]两种物质浓度及孔隙率等因素对于检测结果的影响.结果表明,本研究建立的竞争法模型与仿真能探究各参数对于检测结果的影响,优化试条性能,从而提高试条检测灵敏度和实现定量检测.     

对流扩散方程的高效稳定差分格式

基于二阶修正Dennis格式 ,提出了采用时间相关法求解定常对流扩散方程的一种具有节省内存空间和提高定常解收敛速度的有理式型优化半隐和松驰半隐紧致格式 .本文建立的差分格式具有运算量小、无网格雷诺数限制的优点 ,是无条件稳定和无条件单调的。通过对非线性Burgers方程进行的数值计算结果表明 ,文中构造的有理式型优化半隐和松驰半隐紧致格式适合于非线性问题计算 ,且保持了无条件稳定和无条件单调的特性 ,尤其能使定常解收敛速度加快 ,精度提高 .     

Total variation regularization for the reconstruction of a mountain topography

The aim of this paper is to reconstruct a paleo mountain topography using a total variation (TV) regularization. A coupled system integrates the tectonic process with the surface process to simulate the evolution of a paleo mountain topography. The tectonic process and the surface process are described by a 3D convection-diffusion equation and a 2D convection-diffusion equation, respectively. We recover a piecewise smooth velocity field for the tectonic process as well as reconstruct a piecewise smooth mountain topography for the surface process using a TV regularization in an iterative fashion. The effects of the number of samples and of wavelengths on inversions are investigated. In our numerical experiments, we shall experience three difficulties: (I) recovering a large quantity of information from the limited amount of measurement data; (II) detecting sharp features; (III) choosing a properly initial guess value for a TV regularization. The numerical experiments show that a TV regularization is an efficient and stable algorithm.     

基于多层增量未知元方法的一类三维对流扩散方程的研究

对于一类一般形式的三维对流扩散方程, 运用有限差分方法, 在增量未知元方法(IU)下, 可以得到一个IU型正定但非对称的线性方程组.其系数矩阵条件数要远远优于不用IU方法的情形^[1]. 考虑到IU方法的这一优点, 作者在文中将IU方法与几种经典的迭代方法相结合, 来求解上述系统. 作者从理论上对该系统的IU型系数矩阵条件数进行了估计, 并通过数值试验验证了这几种IU型迭代方法的有效性.     

Local projection stabilisation for higher order discretisations of convection-diffusion problems on Shishkin meshes

We consider a singularly perturbed convection-diffusion equation on the unit square where the solution of the problem exhibits exponential boundary layers. In order to stabilise the discretisation, two techniques are combined: Shishkin meshes are used and the local projection method is applied. For arbitrary r≥2, the standard Q _r -element is enriched by just six additional functions leading to an element which contains the P _r+1. In the local projection norm, the difference between the solution of the stabilised discrete problem and an interpolant of the exact solution is of order uniformly in ε. Furthermore, it is shown that the method converges uniformly in ε of order in the global energy norm.      

A fourth-order method of the convection-diffusion equations with Neumann boundary conditions

In this paper, we have developed a fourth-order compact finite difference scheme for solving the convection-diffusion equation with Neumann boundary conditions. Firstly, we apply the compact finite difference scheme of fourth-order to discrete spatial derivatives at the interior points. Then, we present a new compact finite difference scheme for the boundary points, which is also fourth-order accurate. Finally, we use a Padé approximation method for the resulting linear system of ordinary differential equations. The presented scheme has fifth-order accuracy in the time direction and fourth-order accuracy in the space direction. It is shown through analysis that the scheme is unconditionally stable. Numerical results show that the compact finite difference scheme gives an efficient method for solving the convection-diffusion equations with Neumann boundary conditions.     

Injection profiles in liquid chromatography. I. A fundamental investigation

This is a fundamental experimental and theoretical investigation on how the injection profile depends on important experimental parameters. The experiments revealed that the injection profile becomes more eroded with increased (i) flow rate, (ii) viscosity of the eluent, (iii) size of the solute, (iv) injection volume and (v) inner diameter of the injection loop capillary. These observations cannot be explained by a 1D-convection-diffusion equation, since it does not account for the effect of the parabolic flow and the radial diffusion on the elution profile. Therefore, the 1D model was expanded into a 2D-convection-diffusion equation with cylindrical coordinates, a model that showed a good agreement with the experimental injection profiles dependence on the experimental parameters. For a deeper understanding of the appearance of the injection profile the 2D model is excellent, but to account for injection profiles of various injection volumes and flow rates in preparative and process-chromatography using computer-optimizations, a more pragmatic approach must be developed. The result will give guidelines about how to reduce the extra-column variance caused by the injection profile. This is important both for preparative and analytical chromatography; in particular for modern analytical systems using short and narrow columns.     

Iterative methods for stabilized discrete convection-diffusion problems

In this paper we study the computational cost of solving theconvection-diffusion equation using various discretization strategiesand iteration solution algorithms. The choice of discretizationinfluences the properties of the discrete solution and alsothe choice of solution algorithm. The discretizations consideredhere are stabilized low-order finite element schemes using streamlinediffusion, crosswind diffusion and shock-capturing. The latter,shock-capturing discretizations lead to nonlinear algebraicsystems and require nonlinear algorithms. We compare variouspreconditioned Krylov subspace methods including Newton-Krylovmethods for nonlinear problems, as well as several preconditionersbased on relaxation and incomplete factorization. We find thatalthough enhanced stabilization based on shock-capturing requiresfewer degrees of freedom than linear stabilizations to achievecomparable accuracy, the nonlinear algebraic systems are morecostly to solve than those derived from a judicious combinationof streamline diffusion and crosswind diffusion. Solution algorithmsbased on GMRES with incomplete block-matrix factorization preconditioningare robust and efficient.