Construction of the Local Structure-Preserving Algorithms for the General Multi-Symplectic Hamiltonian System

Many partial differential equations can be written as a multi-symplectic Hamiltonian system, which has three local conservation laws, namely multi-symplectic conservation law, local energy conservation law and local momentum conservation law. In this paper, we systematically give a unified framework to construct the local structure-preserving algorithms for general conservative partial differential equations starting from the multi-symplectic formulation and using the concatenating method. We construct four multi-symplectic algorithms, two local energy-preserving algorithms and two local momentum-preserving algorithms, which are independent of the boundary conditions and can be used to integrate any partial differential equations written in multi-symplectic Hamiltonian form. Among these algorithms, some have been discussed and widely used before while most are novel schemes. These algorithms are illustrated by the nonlinear Schrödinger equation and the Klein-Gordon-Schrödinger equation. Numerical experiments are conducted to show the good performance of the proposed methods.     

Novel Symplectic Discrete Singular Convolution Method for Hamiltonian PDEs

This paper explores the discrete singular convolution method for Hamiltonian PDEs. The differential matrices corresponding to two delta type kernels of the discrete singular convolution are presented analytically, which have the properties of high-order accuracy, band-limited structure and thus can be excellent candidates for the spatial discretizations for Hamiltonian PDEs. Taking the nonlinear Schrödinger equation and the coupled Schrödinger equations for example, we construct two symplectic integrators combining this kind of differential matrices and appropriate symplectic time integrations, which both have been proved to satisfy the square conservation laws. Comprehensive numerical experiments including comparisons with the central finite difference method, the Fourier pseudospectral method, the wavelet collocation method are given to show the advantages of the new type of symplectic integrators.     

Multi-Symplectic Fourier Pseudospectral Method for the Kawahara Equation

In this paper, we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiation matrix and discrete Fourier transform. The relationship is crucial for implementing the scheme efficiently. By using the relationship, we can apply the Fast Fourier transform to solve the Kawahara equation. The effectiveness of the proposed methods will be demonstrated by a number of numerical examples. The numerical results also confirm that the global energy and momentum are well preserved.     

An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient

In this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, the original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization approach. Multiple trial functions are used to enrich the stochastic basis and improve the accuracy. In the online stage, we solve each stochastic subproblem for any given forcing function by projecting the stochastic solution into the data-driven stochastic basis constructed offline. In our ANOVA-DSM framework, solving the original high-dimensional stochastic problem is reduced to solving a series of ANOVA-decomposed stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided to further reduce the number of the stochastic subproblems and speed up our method. To demonstrate the accuracy and efficiency of our method, numerical examples are presented for one- and two-dimensional elliptic PDEs with random coefficients.     

A Kernel-Free Boundary Integral Method for Variable Coefficients Elliptic PDEs

This work proposes a generalized boundary integral method for variable coefficients elliptic partial differential equations (PDEs), including both boundary value and interface problems. The method is kernel-free in the sense that there is no need to know analytical expressions for kernels of the boundary and volume integrals in the solution of boundary integral equations. Evaluation of a boundary or volume integral is replaced with interpolation of a Cartesian grid based solution, which satisfies an equivalent discrete interface problem, while the interface problem is solved by a fast solver in the Cartesian grid. The computational work involved with the generalized boundary integral method is essentially linearly proportional to the number of grid nodes in the domain. This paper gives implementation details for a second-order version of the kernel-free boundary integral method in two space dimensions and presents numerical experiments to demonstrate the efficiency and accuracy of the method for both boundary value and interface problems. The interface problems demonstrated include those with piecewise constant and large-ratio coefficients and the heterogeneous interface problem, where the elliptic PDEs on two sides of the interface are of different types.     

The Bulk-Surface Virtual Element Method for Reaction-Diffusion PDEs: Analysis and Applications

Bulk-surface partial differential equations (BS-PDEs) are prevalent in manyapplications such as cellular, developmental and plant biology as well as in engineering and material sciences. Novel numerical methods for BS-PDEs in three space dimensions (3D) are sparse. In this work, we present a bulk-surface virtual elementmethod (BS-VEM) for bulk-surface reaction-diffusion systems, a form of semilinearparabolic BS-PDEs in 3D. Unlike previous studies in two space dimensions (2D), the3D bulk is approximated with general polyhedra, whose outer faces constitute a flatpolygonal approximation of the surface. For this reason, the method is restricted tothe lowest order case where the geometric error is not dominant. The BS-VEM guarantees all the advantages of polyhedral methods such as easy mesh generation andfast matrix assembly on general geometries. Such advantages are much more relevantthan in 2D. Despite allowing for general polyhedra, general nonlinear reaction kineticsand general surface curvature, the method only relies on nodal values without needing additional evaluations usually associated with the quadrature of general reactionkinetics. This latter is particularly costly in 3D. The BS-VEM as implemented in thisstudy retains optimal convergence of second order in space.     

An Interface-Capturing Method for Resolving Compressible Two-Fluid Flows with General Equation of State

In this study, a stable and robust interface-capturing method is developed to resolve inviscid, compressible two-fluid flows with general equation of state (EOS). The governing equations consist of mass conservation equation for each fluid, momentum and energy equations for mixture and an advection equation for volume fraction of one fluid component. Assumption of pressure equilibrium across an interface is used to close the model system. MUSCL-Hancock scheme is extended to construct input states for Riemann problems, whose solutions are calculated using generalized HLLC approximate Riemann solver. Adaptive mesh refinement (AMR) capability is built into hydrodynamic code. The resulting method has some advantages. First, it is very stable and robust, as the advection equation is handled properly. Second, general equation of state can model more materials than simple EOSs such as ideal and stiffened gas EOSs for example. In addition, AMR enables us to properly resolve flow features at disparate scales. Finally, this method is quite simple, time-efficient and easy to implement.     

Fast Spectral Collocation Method for Surface Integral Equations of Potential Problems in a Spheroid

This paper proposes a new technique to speed up the computation of the matrix of spectral collocation discretizations of surface single and double layer operators over a spheroid. The layer densities are approximated by a spectral expansion of spherical harmonics and the spectral collocation method is then used to solve surface integral equations of potential problems in a spheroid. With the proposed technique, the computation cost of collocation matrix entries is reduced from O(M²N⁴) to O(MN⁴), where N²is the number of spherical harmonics (i.e., size of the matrix) and M is the number of one-dimensional integration quadrature points. Numerical results demonstrate the spectral accuracy of the method.     

An LDG Method for Stochastic Cahn-Hilliard Type Equation Driven by General Multiplicative Noise Involving Second-Order Derivative

In this paper, we propose a local discontinuous Galerkin (LDG) method forthe multi-dimensional stochastic Cahn-Hilliard type equation in a general form, whichinvolves second-order derivative $∆u$ in the multiplicative noise. The stability of ourscheme is proved for arbitrary polygonal domain with triangular meshes. We get thesub-optimal error estimate $mathbb{O}(h^k)$ if the Cartesian meshes with $Q^k$ elements are used.Numerical examples are given to display the performance of the LDG method.     

Conditional Simulation of Flow in Heterogeneous Porous Media with the Probabilistic Collocation Method

A stochastic approach to conditional simulation of flow in randomly heterogeneous media is proposed with the combination of the Karhunen-Loeve expansion and the probabilistic collocation method (PCM). The conditional log hydraulic conductivity field is represented with the Karhunen-Loeve expansion, in terms of some deterministic functions and a set of independent Gaussian random variables. The propagation of uncertainty in the flow simulations is carried out through the PCM, which relies on the efficient polynomial chaos expansion used to represent the flow responses such as the hydraulic head. With the PCM, existing flow simulators can be employed for uncertainty quantification of flow in heterogeneous porous media when direct measurements of hydraulic conductivity are taken into consideration. With illustration of several numerical examples of groundwater flow, this study reveals that the proposed approach is able to accurately quantify uncertainty of the flow responses conditioning on hydraulic conductivity data, while the computational efforts are significantly reduced in comparison to the Monte Carlo simulations.     

介绍一种治疗先天性桡骨缺如的手术方法

目的：报告一种治疗先天性桡骨缺如的手术方法。方法：采用桡侧软组织松解,尺骨远端截骨,“Ｖ”字成型,尺腕关节融合。结果：术后随访８个月,腕手桡偏畸形完全矫正,手指屈伸功能有一定。结论;青少年的损杖手畸形,通过本法矫正,可获满意效果。     

An Old Method for Good New Cells

The aim of this work was to demonstrate a greater number of viable cells using a micro-surgical in-situ perfusion to collect rat pancreata compared with the pancreas after exsanguination. We used 3 groups of 20 rats. Perfusion was performed by selective cannulation of the left common iliac artery with administration of UW solution at 4°C. Collected pancreata were digested and cells separated by Ficoll gradient were placed in culture to permit adhesion to dishes. Cells were characterized and tested for viability. We observed a gain of about 14% in the number of viable cells compared with those obtained after exsanguination (P < .001 by chi-square).     

An Accelerated Method for Simulating Population Dynamics

We present an accelerated method for stochastically simulating the dynamics of heterogeneous cell populations. The algorithm combines a Monte Carlo approach for simulating the biochemical kinetics in single cells with a constant-number Monte Carlo method for simulating the reproductive fitness and the statistical characteristics of growing cell populations. To benchmark accuracy and performance, we compare simulation results with those generated from a previously validated population dynamics algorithm. The comparison demonstrates that the accelerated method accurately simulates population dynamics with significant reductions in runtime under commonly invoked steady-state and symmetric cell division assumptions. Considering the increasing complexity of cell population models, the method is an important addition to the arsenal of existing algorithms for simulating cellular and population dynamics that enables efficient, coarse-grained exploration of parameter space.     

全麻知晓－易勿视的并发症

采用规范化问答对２６４例普鲁卡因－芬太尼全麻病人术后２４ｈ进行知晓调查，其发生率为５．３％。８例知痛，其中１例疼痛严重，并造成术后精神上的创伤，多数发生于缝皮肤的最后几针。本调查范围内的病人知晓发生率明显高于国外全麻报告组，因此必须采用切实可行的预防该并发症的措施。     

压力知觉和积极应对在护生生命意义感与职业认同感间的链式中介作用

阐述了美国麻省总医院的护理管理、临床护理和护理教育的工作方式及内容,认为麻省总医院的护理管理、临床护理和护理教育工作方式对我国护理工作具有指导作用,并指出了我国护理工作的发展现状及方向.     

The Recursive Formulation of Particular Solutions for Some Elliptic PDEs with Polynomial Source Functions

In this paper we develop an efficient meshless method for solving inhomogeneous elliptic partial differential equations. We first approximate the source function by Chebyshev polynomials. We then focus on how to find a polynomial particular solution when the source function is a polynomial. Through the principle of the method of undetermined coefficients and a proper arrangement of the terms for the polynomial particular solution to be determined, the coefficients of the particular solution satisfy a triangular system of linear algebraic equations. Explicit recursive formulas for the coefficients of the particular solutions are derived for different types of elliptic PDEs. The method is further incorporated into the method of fundamental solutions for solving inhomogeneous elliptic PDEs. Numerical results show that our approach is efficient and accurate.