Du Fort-Frankel格式及DFF-I并行格式的稳定性

 本文对抛物型方程的Du Fort-Frankel(DFF)格式以及基于该格式构造的并行差分格式(DFF-I)进行了稳定性分析。采用矩阵分析方法, 证明了其无条件(LR)稳定性, 给出了DFF格式的稳定性系数的最小值的上界估计, 结果表明其与网格比有关, 从而DFF格式并非绝对稳定。本文改进了并行差分格式(DFF-I)的稳定性分析结果, 证明了其增长矩阵的谱半径严格小于1, 从而具有长时间稳定性。数值算例验证了DFF-I格式具有空间二阶精度, 且有很好的稳定性。     

广义sine-Gordon方程高精度隐式紧致差分方法

本文研究一类二维非线性的广义sine-Gordon(简称SG)方程的有限差分格式.首先构造三层时间的紧致交替方向隐式差分格式,并用能量分析法证明格式具有二阶时间精度和四阶空间精度.然后应用改进的Richardson外推算法将时间精度提高到四阶.最后,数值算例证实改进后的算法在空间和时间上均达到四阶精度.     

三维多面体网格上扩散方程的保正格式

 针对三维任意(星形)多面体网格, 本文构造了扩散方程的一种单元中心型非线性有限体积格式, 证明了该格式具有保正性. 在该格式设计中, 除引入网格中心量外, 还引入网格节点量和网格面中心量作为中间未知量, 它们将用网格中心未知量线性组合表示, 使得格式仅有网格中心未知量作为基本未知量. 在节点量计算中, 利用网格面上的调和平均点, 设计了一种适用于三维多面体网格的局部显式加权方法. 该格式适用于求解非平面的网格表面和间断扩散系数的问题. 数值例子验证了它对光滑解具有二阶精度和保正性.     

基于深度学习的“实验方案设计”主题式复习教学——以“阿司匹林的制备和提纯”为例

针对实验方案设计题的得分率分析和教学现状,基于课程标准、教材、真题和文献研究,构建“阿司匹林的制备和提纯”深度学习主题模型。通过设计制备和提纯阿司匹林的实验方案,构建实验方案设计的思维模型,并以制备七水合硫酸锌为例内化思维模型,在解决真实、复杂的问题过程中实现核心知识结构化,生成指向深度学习的主题式复习教学流程。     

基于“模型认知”的定量实验方案设计的教学实践*——摩尔盐纯度的测定

针对高三定量实验的复习特点,以“摩尔盐纯度的测定”为例进行基于“模型认知”核心素养培养的教学实践。对教学背景、设计思路、教学过程等方面进行研究分析,并从精心选素材,引发模型认知;高质量设问,建构模型认知;点、线、面结合,丰富模型认知等3个角度对促进学生“模型认知”素养提升的化学教学进行分析和阐释。     

Adaptive discretization for signal detection in statistical inverse problems

We discuss statistical tests in inverse problems when the original equation is replaced by a discretized one, i.e. a linear system of equations. Previous studies revealed that using the discretization level as regularizing procedure is possible, but its application is limited unless discretization is restricted to the singular value decomposition, see C. Marteau and P. Mathé, General regularization schemes for signal detection in inverse problems, 2013. General linear regularization may circumvent this, and we propose a regularization of the discretized equations. The discretization level may be chosen adaptively, which may save computational budget. This results in tests which are known to yield the optimal separation rate up to some constant in many cases.     

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015     

Transient pockets as mediators of gas molecules routes inside proteins: The case study of dioxygen pathway in homogentisate 1,2-dioxygenase and its implication in Alkaptonuria development

Alkaptonuria (AKU) is an ultra-rare disease caused by mutations in homogentisate 1,2-dioxygenase (HGD) enzyme, characterized by the loss of enzymatic activity and the accumulation of its substrate, homogentisic acid (HGA) in different tissues, leading to ochronosis and organ degeneration. Although the pathological effects of HGD mutations are largely studied, less is known about the structure of the enzyme, in particular the pathways for dioxygen diffusion to the active site, required for the enzymatic reaction, are still uninvestigated. In the present project, the combination of two in silico techniques, Molecular Dynamics (MD) simulation and Implicit Ligand Sampling (ILS), was used to delineate gas diffusion routes in HGD enzyme. A route from the central opening of the hexameric structure of the enzyme to the back of the active site trough the protein moiety was identified as the path for dioxygen diffusion, also overlapping with a transient pocket, which then assumes an important role in dioxygen diffusion. Along the route the sequence location of the missense variant E401Q, responsible for AKU development, was also found, suggesting such mutation to be conducive of enzymatic activity loss by altering the flow dynamics of dioxygen. Our in silico approach allowed also to delineate the route of HGA substrate to the active site, until now only supposed.     

A cell functional minimization scheme for parabolic problem

We are interested in a robust and accurate finite volume scheme for 2-D parabolic problems derived from the cell functional minimization approach. The scheme has a local stencil, is locally conservative, treats discontinuity rigorously and leads to a symmetric positive definite linear system. Since the scheme has both cell centered unknowns and cell edge unknowns, the computational cost is an issue and a parallel algorithm is then suggested based on nonoverlapping domain decomposition approach. The interface condition is of the Dirichlet–Robin type and has a parameter λ. By choosing this parameter properly, the convergence of the iteration process could be sped up. Numerical results for linear and nonlinear problems demonstrate the good performance of the cell functional minimization scheme and its parallel version on distorted meshes.     

A class of deformational flow test cases for linear transport problems on the sphere

A class of new benchmark deformational flow test cases for the two-dimensional horizontal linear transport problems on the sphere is proposed. The scalar field follows complex trajectories and undergoes severe deformation during the simulation; however, the flow reverses its course at half-time and the scalar field returns to its initial position and shape. This process makes the exact solution available at the end of the simulation, and facilitates assessment of the accuracy of the underlying transport scheme. A procedure to eliminate possible cancellations of errors when the flow reverses is proposed.