基于一维移动物体的双目装置自标定

给出了平移运动的一维物体所在平面的虚圆点图像及其对摄像机内参数的约束，和约束方程的数值求解方法，从而获得摄像机的内参数. 进一步通过恢复空间点在摄像机坐标系中的坐标，求解出双目摄像机之间的方位，即摄像机的外参数．对于一维物体的一般刚体运动，给出了把它转化为平移运动的方法．模拟实验和真实图像实验结果表明该方法具有较高的求解精度，同时也有一定的应用价值．     

三维非定常不可压涡量—速度Navier—Stokes方程组的有限差分法

提出了一种数值求解三维非定常涡量—速度形式的不可压Navier-Stokes方程组的有限差分方法,该方法在空间方向上具有二阶精度,并且系数矩阵具有对角占优性,因此适合高雷诺数问题的数值求解．同时,给出了适合的二阶涡量边界条件．通过对有精确解的狄利克雷边值问题和典型的驱动方腔流问题的数值实验,验证了本文格式的精确性、稳定性和有效性．     

一种求解混合约束问题的快速完备算法

布尔与数值变量相混合的约束问题有着广泛盼应用，但是当约束中的数值变量间存在非线性关系时该问题求解起来十分困难．目前的许多求解方法都是不完备的，即这些方法不能完全肯定某些包含非线性数值表达式的约束是否能够成立．针对这种问题，提出了数值与区间分析相结合进行数值约束求解的方法．已经实现了一个基于此方法的原型工具．实验结果表明。该方法能够有效、快速、完备地求解非线性混合约束问题．     

泰勒有限差分方法数值求解常微分方程

本文提出一种新的更高精度的泰勒有限差分公式并且应用于求解常微分方程。这种应用泰勒有限差分公式来求解常微分方程的方法称为泰勒有限差分方法。此外,出于比较的目的,使用欧拉方法求解常微分方程的算法流程也被提及,并且在MATLAB软件平台进行了两组对比性的数值实验。两组对比性的数值实验结果均表明,使用泰勒有限差分方法求解常微分方程的精度要比使用欧拉方法求解常微分方程的精度更高。后续可以应用该方法去开发常微分方程数值求解器的软件。     

半解析高阶有限谱元法及其在波导介质层PBG结构滤波器优化设计中的应用

本文采用高阶有限谱单元对分层结构的横截面进行半解析离散．将结构中沿纵向均匀的区段视为子结构，运用基于Riccati方程的精细积分算法求出其出口刚度阵．网格拼装后即可对分层介质问题进行求解．半解析高阶谱单元的采用可以避免著名的龙格现象，该算法的数值精度能随着基函数的阶数的增加呈指数级提高．即高阶有限谱单元能够达到任意需要的精度．数值算例证明这种方法具有很高的精度与效率．在高精度高效率分析的基础上建立了滤波器的优化设计模型，利用遗传算法对优化模型进行全局优化，得到了PBG结构滤波性能全局最优的设计参数．     

重叠组合法的芯片级三维寄生电容提取及其并行实现

采用双向区域重叠组合法，基于三维层次式块边界元法实现了芯片级的互连电容提取．该方法将芯片切分为大量小规模区域。用全局场求解器计算各子区域电容矩阵，可方便地组合出整个芯片的电容矩阵；同时分析了其计算量和精度，并进行了并行计算实验．对实际版图结构的数值实验验证了有关分析结论，表明该方法高效、可靠、并行性能好．     

求解常微分方程初值问题的并行块预估——校正方法

针对多处理机系统构造了一类具有较高并行度的并行块预估－校正方法。在ｋ＝２，ｓ＝３的情况下，给出了一个具有四介精度的并行计算公式，并讨论了该方法的稳定性，数值结果表明该计算公式对求解常微分方程是有效的。ｒ     

二维波动方程的一种高精度紧致差分方法

提出了一种求解二维波动方程的高精度紧致差分方法,该方法首先利用紧交替方向隐式差分格式,其截断误差为O(τ2+h4),分别在粗网格和细网格上对原方程进行求解,然后利用Richardson外推计算一次,进一步提高精度,得到了二维波动方程具有O(τ4+h6)精度的数值解。数值实验验证了该方法的可靠性、有效性和精确性。     

求解双曲守恒律及其对流扩散方程的中心迎风方法

提出了一种新的求解双曲守恒律方程(组)的四阶半离散中心迎风差分方法．空间导数项的离散采用四阶CWENO(central weighted essentially non—oscillatory)的构造方法,使所得到的新方法在提高精度的同时,具有更高的分辨率．使用该方法产生的数值粘性要比交错的中心格式小,而且由于数值粘性与时间步长无关,从而时间步长可根据稳定性需要尽可能的小．     

二维Helmholtz方程外问题的数值解法

利用位势理论把Helmholtz方程外问题转化为第二类积分方程的求解问题．在处理积分算子核时,采用了一种新的裂解方式,再利用Nystrom方法求解数值结果．最后针对该方法给出数值实例,以表明此方法的有效性．     

Multi-dimensional semi-Lagrangian scheme that guarantees exact conservation

A new numerical method that guarantees exact mass conservation is proposed to solve multi-dimensional hyperbolic equations in semi-Lagrangian form without directional splitting. The method is based on a concept of CIP scheme and keep the many good characteristics of the original CIP scheme. The CIP strategy is applied to the integral form of variable. Although the advection and non-advection terms are separately treated, the mass conservation is kept in a form of spatial profile inside a grid cell. Therefore, it retains various advantages of the semi-Lagrangian schemes with exact conservation that has been beyond the capability of conventional semi-Lagrangian schemes.     

基于流体动力学的火焰实时绘制技术

为了实时有效地渲染真实的火焰，引入了基于流体动力学的气体建模方法和分形插值技术，在粗网格下采用半拉格朗日方法和隐式差分格式，直接求取火焰的纳威-斯托克斯方程，这种数值解法在粗糙网格、大的时间步下也能无条件稳定，能达到实时渲染的目的。在细网格下，为了渲染湍流火焰，采用分形插值的方法，增强湍流火焰的边缘细节。实验结果表明：该方法实现简单，仿真速度快，显示的动画效果真实，并且是元条件稳定的。     

Fluid simulation with adaptively sharpening and embedded boundary conditions

In this paper, we present a physically based technique for simulating inviscid fluids. Our contribution is concerned with two issues. First, for solving the advection equation, we introduce a hybrid scheme that couples the FLIP scheme with the semi-Lagrangian scheme by adaptively distributing implicit particles and using a transition layer to propagate information. Secondly, for solving pressure, we develop a flux based scheme that can embed arbitrary solid boundaries into a Poisson equation. And based on this scheme we make further improvement to achieve two-way fluid/solid coupling on an octree structure with second-order accuracy. Finally, the experimental results demonstrate that our hybrid scheme for advection can preserve relatively fine surface details with less computation expenditure; and simultaneously our robust pressure solver can handle both stationary and moving obstacles more efficiently compared with unstructured meshes.     

Large eddy simulation of turbulent heat transport in the Strait of Gibraltar

We develop a numerical model for large eddy simulation of turbulent heat transport in the Strait of Gibraltar. The flow equations are the incompressible Navier–Stokes equations including Coriolis forces and density variation through the Boussinesq approximation. The turbulence effects are incorporated in the system by considering the Smagorinsky model. As a numerical solver we propose a finite element semi-Lagrangian method. The solution procedure consists of combining a non-oscillatory semi-Lagrangian scheme for time discretization with the finite element method for space discretization. Numerical results illustrate a buoyancy-driven circulations along the Strait of Gibraltar and the sea-surface temperature is flushed out and move to northeast coast. The Ocean discharge and the temperature difference are shown to control the plume structure.     

Exact Boundary Condition for Semi-discretized Schrödinger Equation and Heat Equation in a Rectangular Domain

An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function. The value function can be characterized as the solution of an evolutionary Hamilton–Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored. The approach is illustrated for the wave equation and an extension to equations of Schrödinger type is indicated. We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory.     

Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

We present a review of the semi-Lagrangian method for advection-diffusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable.     

High Order Multi-dimensional Characteristics Tracing for the Incompressible Euler Equation and the Guiding-Center Vlasov Equation

In this paper, we propose a high order characteristics tracing scheme for the two-dimensional nonlinear incompressible Euler system in vorticity stream function formulation and the guiding center Vlasov model. Such a scheme is incorporated into a semi-Lagrangian finite difference WENO framework for simulating the aforementioned model equations. This is an extension of our earlier work on high order characteristics tracing scheme for the 1D nonlinear Vlasov–Poisson system (Qiu and Russo in J Sci Comput 71:414–434, 2017). The effectiveness of the proposed scheme is demonstrated numerically by an extensive set of test cases.     

Instability in leapfrog and forward–backward schemes: Part II: Numerical simulations of dam break

Following Sun’s approach [17], Shuman smoothing instead of conventional diffusion terms is used in a simple two-time step semi-implicit finite volume scheme to simulate dam break. When the Courant number is less than one, the absolute value of amplification factor of the 1D linearized shallow-water equations is 1 in this new scheme. Compared with the characteristic-based semi-Lagrangian schemes and the Riemann solver, this scheme produces excellent results of free water depth and speed of the shock. Numerical simulations show that the water inside the dam initially moves away radially until water almost depletes near the center; then the water moves back to the center and forms a vertical water column there. This paper proves that Shuman smoothing can be used not only in the linearized shallow-water equations discussed in Sun [17] but also in the nonlinear wave equations to control instability around shocks.     

An Adaptive Sparse Grid Semi-Lagrangian Scheme for First Order Hamilton-Jacobi Bellman Equations

We propose a semi-Lagrangian scheme using a spatially adaptive sparse grid to deal with non-linear time-dependent Hamilton-Jacobi Bellman equations. We focus in particular on front propagation models in higher dimensions which are related to control problems. We test the numerical efficiency of the method on several benchmark problems up to space dimension d=8, and give evidence of convergence towards the exact viscosity solution. In addition, we study how the complexity and precision scale with the dimension of the problem.     

Strong and Auxiliary Forms of the Semi-Lagrangian Method for Incompressible Flows

We present a review of the semi-Lagrangian method for advection–diffusion and incompressible Navier–Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable.