Three dimensional discontinuous Galerkin methods for Euler equations on adaptive conforming meshes

In the numerical simulation of three dimensional fluid dynamical equations, the huge computational quantity is a main challenge. In this paper, the discontinuous Galerkin (DG) finite element method combined with the adaptive mesh refinement (AMR) is studied to solve the three dimensional Euler equations based on conforming unstructured tetrahedron meshes, that is according the equation solution variation to refine and coarsen grids so as to decrease total mesh number. The four space adaptive strategies are given and analyzed their advantages and disadvantages. The numerical examples show the validity of our methods.     

A Newton-like finite element scheme for compressible gas flows

Semi-implicit and Newton-like finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a high-resolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVD-type flux limiters. A semi-implicit scheme is derived by a time-lagged linearization of the nonlinear residual, and a Newton-like method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. Numerical evidence for unconditional stability is presented. It is shown that the proposed approach offers higher accuracy and better convergence behavior than algorithms in which the boundary conditions are implemented in a strong sense.     

Approximating Euclidean circles by neighbourhood sequences in a hexagonal grid

In this paper the nodes of the hexagonal grid are used as points. There are three types of neighbours on this grid, therefore neighbourhood sequences contain values 1, 2, 3. The grid is coordinatized by three coordinates in a symmetric way. Digital circles are classified based on digital distances using neighbourhood sequences. They can be triangle, hexagon, enneagon and dodecagon. Their corners and side-lengths are computed, such as their perimeters and areas. The radius of a digital disk is usually not well-defined, i.e., the same disk can have various radii according to the neighbourhood sequence used. Therefore the non-compactness ratio is used to measure the quality of approximation of the Euclidean circles. The best approximating neighbourhood sequence is presented. It is shown that the approximation can be improved using two neighbourhood sequences in parallel. Comparisons to other approximations are also shown.     

三维超音进气道系统流场计算的区域分裂算法

51．引言对于现代战斗机和飞航导弹来说,进气道的气流品质对推进系统的性能有很大的影响．进气道的气流品质不仅依赖于进气道的形状,也依赖于飞行器的前部外型和进气道的安装位置．因此,必须对前机（弹）身外流场和进气道内流场统一求解,以适应一体化设计之要求．然而统一求解三推进气道内外流场组合系统有许多困难．首先是难以生成单一的计算网格系统,其次是在很大程度上受计算机内存和运算速度的限制．统一的网格系统必然需要很大的内存．另外,由于进气道内外流场相互干扰十分强烈,在某些来流条件下,一般的统一网格极难得到收敛解…     

水流动画模拟方法探讨

目前,对于水流动画模拟的研究已经取得了相当丰富的成果。基于物理模型的流体动画模拟的研究,需要计算流体力学和计算机图形学的交叉融合,根据其研究的背景与内容的不同,可分为两种类型。基于物理模型的流体动画模拟中描述流体运动的方法主要有两种,一种是Euler方法,另一种是Lagrange方法。Euler方法的主要缺点是难以处理流体的细节,Lagrange方法的优点就是能很好地表现流体的细节。由于Euler方法和Lagrange方法的这些特点,如果发展这两种方法的综合方法,则可取长补短。     

Random Sampling of Euler Tours

We define a Markov chain on the set of Euler tours of a given Eulerian graph based on transformations first defined by Kotzig in 1966. We prove that the chain is rapidly mixing if the maximum degree in the given graph is 6, thus obtaining an efficient algorithm for sampling and counting the set of Euler tours for such an Eulerian graph. Received October 30, 1997; revised March 12, 1999, and April 17, 2000.     

Time stepping in discontinuous Galerkin method

The time discretization in the Discontinuous Galerkin (DG) scheme has been traditionally based on the Total Variation Diminishing (TVD) second-order Runge-Kutta (RK2) scheme. Computational efficiency and accuracy with the Euler Forward (EF) and the TVD second-order RK2 time stepping schemes in the DG method are investigated in this work. Numerical tests are conducted with the scalar Burgers equation, 1-D and 2-D shallow water flow equations. The maximum Courant number or time step size required for stability for the EF scheme and RK2 scheme with different slope limiters are compared. Numerical results show that the slope limiters affect the stability requirement in the DG method. The RK2 scheme is generally more diffusive than the EF scheme, and the RK2 scheme allows larger time step sizes. The EF scheme is found to be more efficient and accurate than the RK2 scheme in the DG method in computation.     

Delay-dependent exponential stability of the backward Euler method for nonlinear stochastic delay differential equations

Recently, several scholars discussed the question of under what conditions numerical solutions can reproduce exponential stability of exact solutions to stochastic delay differential equations, and some delay-independent stability criteria were obtained. This paper is concerned with delay-dependent stability of numerical solutions. Under a delay-dependent condition for the stability of the exact solution, it is proved that the backward Euler method is mean-square exponentially stable for all positive stepsizes. Numerical experiments are given to confirm the theoretical results.     

Optimal error estimates of the penalty method for the linearized viscoelastic flows

In this article, optimal error estimates of the penalty method for the linearized viscoelastic flows equations arising in the Oldroyd model are derived. Furthermore, error estimates for the backward Euler time discretization scheme in L ² and H ¹-norms are obtained.     

Image processing applications via time-multiplexing cellular neural network simulator with numerical integration algorithms

A novel approach to simulate cellular neural networks (CNN) is presented in this paper. The approach, time-multiplexing simulation, is prompted by the need to simulate hardware models and test hardware implementations of CNN. For practical applications, due to hardware limitations, it is impossible to have a one-to-one mapping between the CNN hardware processors and all the pixels of the image. This simulator provides a solution by processing the input image block by block, with the number of pixels in a block being the same as the number of CNN processors in the hardware. The algorithm for implementing this simulator is presented along with popular numerical integration algorithms. Some simulation results and comparisons are also presented.