一类带扰动的Sine-Gordon方程的谱方法

研究一类具扰动的Sine-Gordon方程u_tt-u_xx+αsinu-βu_xxt=g(u),t>0,-∞< x< ∞的周期初值问题,提出了谱方法,并用先验估计方法作了误差估计,证明了近似方法的收敛性,并得到了该问题广义解的存在、唯一性。     

一类广义强阻尼Sine-Gordon方程的整体解

同时考虑黏性效应及外阻尼作用研究了一类广义强阻尼Sine-Gordon方程-利用Galerkin方法,首先证明了该方程在初值u(x,0)∈H¹₀(Ω),u_t(x,0)∈L²(Ω)的条件下初边值问题存在整体弱解u(x,t),并证明了整体弱解关于初始条件具有 关键词： Sine-Gordon型方程 强阻尼 Galerkin方法 整体解     

带源项的变系数非线性反应扩散方程的精确解

本文利用广义条件对称方法对带源项的变系数非线性反应扩散方程 f(x)u_t=(g(x)D(u)u_x)_x+h(x)P(u)u_x+q(x)Q(u)进行研究. 当扩散项D(u)取u^m (m≠-1,0,1)和e^u两种重要情形时, 对该方程进行对称约化,得到了具有广义泛函分离变量形式的精确解. 这些精确解包含了该方程对应常系数情况下的解. 关键词： 广义条件对称 精确解 非线性反应扩散方程     

Stokes问题的非协调广义差分法

本文提出解Stokes问题的非协调广义差分法,速度用不连续分片线性函数逼近,压力用分片常数函数逼近。速度近似按‖.‖_h模,压力近似按L²模都具有最优的逼近阶误差估计。本文还给出了数值计算结果。     

一类演化方程的高稳定性的差分格式

考虑一类演化方程u_t=au∂_2k+1(其中a是常数,u∂_2k+1=∂^2k+1u/∂x^2k+1,k=1,2……)的有限差分解法。构造了两类具有高稳定性的显式差分格式。并用引入耗散项的方法建立了两类半显式差分格式,它们是无条件稳定的且可显式地进行计算。     

生物组织的δ-P1近似漫反射光学模型

本文研究了生物组织的改进的δ-P₁近似漫反射光学模型,推导了含有等效光源一阶矩的双点源近似空间分辨漫反射解R_δ－P1(ρ).研究表明,考虑等效光源一阶矩的光学模型,较好地描述了具有强的前向散射特性和较大吸收系数的生物组织散射特性;与漫射近似下的漫反射率R_SDA(ρ)相比,新的光学模型能较好地描述光源附近的漫辐射强度分布,并且由于解析表达式中含有散射相函数的二阶参量γ,这对 关键词： 组织光学 1近似')" href="#">δ-P₁近似 等效光源 微区漫反射     

势问题的无单元Galerkin方法的误差估计

在高维情况下,首先研究了无单元Galerkin方法的形函数构造方法——移动最小二乘法在Sobolev空间W^k,p(Ω)中的误差估计.然后,在势问题的无单元Galerkin方法的基础上,研究了势问题的通过罚函数法施加本质边界条件的无单元Galerkin方法在Sobolev空间中的误差估计.当节点和形函数满足一定条件时,证明了该误差估计是最优阶的.从误差分析中可以看出,数值解的误差与权函数的影响半径密切相关.最后,通过算例验证了结论的正确性. 关键词： 无网格方法 无单元Galerkin方法 势问题 误差估计     

W22[a,b]空间中的最佳Hermite插值算子

本文介绍具有再生核的函数Hilbere空间W₂²给出了W₂²空间再生核的有限表达式,利用它构造出最佳Hilbere插值逼近算子(H_2nu)(x)的真体表达式,当节点系无限加密时,能够保证(H_2nu)(x)一致收敛于u(x),(H_2nu)(y)一致收敛于u'(x),且每增加一个节点,误差在Sobolev范数意义下单调下降。     

色散方程的一类具任意稳定性的显格式

本文对色散方程u_r=au_xxx构造了一类中间层包含四个结点,带两个参数m和θ的三层显式差分格式。当m和θ满足一定的关系时,其稳定性条件为|γ|≤(m+1)/(4(m-1))(|m|>1),从而当取m充分接近1时,可得到任意大的稳定性条件,并且保持截断误差阶不变。数值例子验证了理论分析的结果。     

扩散序谱(DOSY)实验测定缓冲体系中蛋白质表观分子量

液体核磁共振扩散序谱（DOSY）可以通过测定溶质分子的自扩散系数（D_t）来研究该分子在溶液中的表观分子量（M）.D_t与测试体系和分子本身性质相关，蛋白质体系较为复杂，从而增加了蛋白质自扩散系数（D_t-protein）测定的难度.本文以3-（三甲基硅基）丙磺酸钠（DSS）为内标，以蛋白质分子与DSS自扩散系数的比值（D_r）来表征蛋白质分子在溶液中的表观分子量（M_protein），该方法降低了缓冲体系对M_protein的影响，使得M_protein主要由分子本身的性质决定.在此基础上，测定了不同分子量蛋白质分子相对于DSS的D_r，拟合得到了D_r与M_protein的相关关系：lgM_protein =-2.6488 lgD_r-0.7863，相关系数（R²）为0.997.最后测定了通过大肠杆菌表达纯化得到的SARS冠状病毒主蛋白酶C端结构域（M^pro-C）分子相对于DSS的D_r，并计算出与文献结果一致的M_protein，进一步验证了拟合公式的准确性和实用性.     

Minimization of the Truncation Error by Grid Adaptation

A new grid adaptation strategy, which minimizes the truncation error of a pth-order finite difference approximation, is proposed. The main idea of the method is based on the observation that the global truncation error associated with discretization on nonuniform meshes can be minimized if the interior grid points are redistributed in an optimal sequence. The method does not explicitly require the truncation error estimate, and at the same time, it allows one to increase the design order of approximation globally by one, so that the same finite difference operator reveals superconvergence properties on the optimal grid. Another very important characteristic of the method is that if the differential operator and the metric coefficients are evaluated identically by some hybrid approximation, then the single optimal grid generator can be employed in the entire computational domain independently of points where the hybrid discretization switches from one approximation to another. Generalization of the present method to multiple dimensions is presented. Numerical calculations of several one-dimensional and one two-dimensional test examples demonstrate the performance of the method and corroborate the theoretical results.     

Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D poisson equation

We develop a sixth order finite difference discretization strategy to solve the two dimensional Poisson equation, which is based on the fourth order compact discretization, multigrid method, Richardson extrapolation technique, and an operator based interpolation scheme. We use multigrid V-Cycle procedure to build our multiscale multigrid algorithm, which is similar to the full multigrid method (FMG). The multigrid computation yields fourth order accurate solution on both the fine grid and the coarse grid. A sixth order accurate coarse grid solution is computed by using the Richardson extrapolation technique. Then we apply our operator based interpolation scheme to compute sixth order accurate solution on the fine grid. Numerical experiments are conducted to show the solution accuracy and the computational efficiency of our new method, compared to Sun–Zhang’s sixth order Richardson extrapolation compact (REC) discretization strategy using Alternating Direction Implicit (ADI) method and the standard fourth order compact difference (FOC) scheme using a multigrid method.     

Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling in three steps.First, a new a priori error estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. This estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. It involves the interpolation error of the Euler fluxes weighted by the gradient of the adjoint state associated with the observed functional. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal continuous mesh is then derived analytically. Thus, it can be used as a metric tensor field to drive the mesh adaptation. From a numerical point of view, this method is completely automatic, intrinsically anisotropic, and does not depend on any a priori choice of variables to perform the adaptation.3D examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency.     

An optimized spectral difference scheme for CAA problems

In the implementation of spectral difference (SD) method, the conserved variables at the flux points are calculated from the solution points using extrapolation or interpolation schemes. The errors incurred in using extrapolation and interpolation would result in instability. On the other hand, the difference between the left and right conserved variables at the edge interface will introduce dissipation to the SD method when applying a Riemann solver to compute the flux at the element interface. In this paper, an optimization of the extrapolation and interpolation schemes for the fourth order SD method on quadrilateral element is carried out in the wavenumber space through minimizing their dispersion error over a selected band of wavenumbers. The optimized coefficients of the extrapolation and interpolation are presented. And the dispersion error of the original and optimized schemes is plotted and compared. An improvement of the dispersion error over the resolvable wavenumber range of SD method is obtained. The stability of the optimized fourth order SD scheme is analyzed. It is found that the stability of the 4th order scheme with Chebyshev–Gauss–Lobatto flux points, which is originally weakly unstable, has been improved through the optimization. The weak instability is eliminated completely if an additional second order filter is applied on selected flux points. One and two dimensional linear wave propagation analyses are carried out for the optimized scheme. It is found that in the resolvable wavenumber range the new SD scheme is less dispersive and less dissipative than the original scheme, and the new scheme is less anisotropic for 2D wave propagation. The optimized SD solver is validated with four computational aeroacoustics (CAA) workshop benchmark problems. The numerical results with optimized schemes agree much better with the analytical data than those with the original schemes.     

Three-dimensional myocardial strain analysis based on short- and long-axis magnetic resonance tagged images using a 1D displacement field

A robust algorithm to estimate three-dimensional strain in the left-ventricular heart wall, based on magnetic resonance (MR) grid-tagging in two sets of orthogonal image planes, is presented. Starting-point of this study was to minimize global interpolation and smoothing. Only the longitudinal displacement was interpolated between long-axis images. Homogeneous strain analysis was performed using small tetrahedrons. The method was tested using a stack of short-axis images and three long-axis images in six healthy volunteers. In addition, the method was subjected to an analytical test case, in which the effect of noise in tag point position on the observed strains was explored for normally distributed noise (0.5 mm RMS). In volunteers, the error in the longitudinal displacement due to interpolation between the long-axis image planes was -0.10 +/- 0. 48 mm (mean +/- SD). The resulting error in the longitudinal strain epsilon(l) was -0.003 +/- 0.02. The analytical test case was used to quantify the effects of three sources of errors on the observed strain. The SD of the difference between homogeneous strain and true strain was 0.06 for epsilon(r.) The error due to the 3-D reconstruction was 0.004 for epsilon(r.) The error in epsilon(r) resulting from simulated noise in the tag point position was 0.10. Equivalent results were obtained for all other strain parameters; thus, the error resulting from noise in the tag point position dominates the error introduced by approximations in the method. Because the proposed method uses a minimum of global interpolation and smoothing, it offers the prospect to detect small regions of aberrant contraction.     

Optimal constant shape parameter for multiquadric based RBF-FD method

Radial basis functions (RBFs) have become a popular method for interpolation and solution of partial differential equations (PDEs). Many types of RBFs used in these problems contain a shape parameter, and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter. In this paper, we focus on PDE problems solved with a multiquadric based RBF finite difference (RBF-FD) method. We propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error. The algorithm is based on analytical approximations to the local RBF-FD error derived in [1]. We show through several examples in 1D and 2D, both with structured and unstructured nodes, that very accurate solutions (compared to finite differences) can be achieved using the optimal value of the constant shape parameter.     

Multi-dimensional conservative semi-Lagrangian method of characteristics CIP for the shallow water equations

A new characteristic approach that guarantees conservative property is proposed and is applied to the shallow water equations. CIP–CSL (Constrained Interpolation Profile/Conservative Semi-Lagrangian) interpolation is applied to the CIP method of characteristics in order to enhance the mass conservation of the numerical result. Although the characteristic formulation is originally derived from non-conservative form, present scheme achieves complete mass conservation by solving mass conservation simultaneously and reflecting conserving mass in interpolation profile. Present method has less height error compared to the CIP method of characteristics by several orders of magnitude. By the enhanced conservation property, present scheme is applicable to nonlinear problem such as shock. Furthermore, application to two dimensions including the Coriolis term is straightforward with directional splitting technique.     

Standard and goal-oriented adaptive mesh refinement applied to radiation transport on 2D unstructured triangular meshes

Standard and goal-oriented adaptive mesh refinement (AMR) techniques are presented for the linear Boltzmann transport equation. A posteriori error estimates are employed to drive the AMR process and are based on angular-moment information rather than on directional information, leading to direction-independent adapted meshes. An error estimate based on a two-mesh approach and a jump-based error indicator are compared for various test problems. In addition to the standard AMR approach, where the global error in the solution is diminished, a goal-oriented AMR procedure is devised and aims at reducing the error in user-specified quantities of interest. The quantities of interest are functionals of the solution and may include, for instance, point-wise flux values or average reaction rates in a subdomain. A high-order (up to order 4) Discontinuous Galerkin technique with standard upwinding is employed for the spatial discretization; the discrete ordinates method is used to treat the angular variable.     

基于遗传算法的传声器阵列位置有源校正研究

传声器阵列的位置误差会导致高精度定位应用场景中算法性能下降,为解决这一问题,该文提出了一种二维平面传声器阵列位置参数的有源校正方法.在校正声源位置已知的情况下,利用各阵元间的到达时间差均方误差之和最小化作为优化目标设计代价函数,根据遗传算法搜索阵列真实位置的全局最优解,从而估计得到传声器阵列的准确位置信息.仿真与半消声...     

基于GF-4PMI数据的亮温差校正火点检测方法研究

高分四号PMI可为防灾减灾提供稳定数据,其搭载的中红外传感器可以很好地应用于快速火灾监测中。但由于缺少传统火灾监测的热红外波段,高分四号提供的光谱信息大多作为灾中监测的辅助数据,且现有的火点识别研究所构建的火点自适应阈值检测算法受单一波段的影响,错检率和漏检率均偏高。为进一步探究高分四号数据在林火监测中的应用方法,提高火点识别精度,本研究分析高分四号数据的特点,结合单通道红外光谱的火点监测方法,应用上下文思想提出一种基于双时相影像的亮温差校正火点检测的方法来进一步提高检测精度。该方法使用灾前和灾中两期影像,具体分为时间尺度上基于空间插值的亮温补偿获取,空间尺度上的上下文自适应阈值分割以及火点判识三个部分。首先将两期影像做差值处理,并将潜在火点周围动态邻域内其他无污染像元的亮温差作为采样点进行空间插值,随后将插值结果带入灾前影像中得到灾中未发生火灾时的背景亮温,最后利用判别条件进行火点判别和虚警剔除,得到最终火点检测结果。其中在灾中背景亮温的预测研究对比了反距离加权插值(inverse distance weigh)、简单克里金插值(simple kriging)和普通克里金插值(ordinary kriging)三种插值方法,从拟合结果来看普通克里金插值既体现了像素区域的波动性又有一定的平滑效果避免峰值过高,是较为理想的拟合结果。实验以目视解译的火点数据为参照验证了山西沁源县和内蒙古呼伦贝尔新巴尔虎左旗地区的两起火灾,对比最新提出的单时相火点检测算法,研究结果表明引进的亮温差校正数据可以更好地拟合背景亮温,减少错分误差至3%,并保持综合评价指标F_β分数在0.9以上。该方法有效结合了高分四号空间和时间的信息,未来可用于高分四号PMI数据自动化火点检测与快速提取。