A numerical study of flow over a confined backward-facing step

Numerical solutions using the SIMPLE algorithms for laminar flow over a backward-facing step are presented. Five differencing schemes were used: hybrid; quadratic upwind (QUICK); second-order upwind (SOUD); central-differencing and a novel scheme named second-order upwind biased (SOUBD). The SOUBD scheme is shown to be part of a family of schemes which include the central-differencing, SOUD and QUICK schemes for uniform grids. The results of the backward-facing step problem are presented and are compared with other numerical solutions and experimental data to evaluate the accuracy of the differencing schemes. The accuracy of the differencing schemes was ascertained by using uniform grids of various grid densities. The QUICK, SOUBD and SOUD schemes gave very similar accurate results. The hybrid scheme suffered from excessive diffusion except for the finest grids and the central-differencing scheme only converged for the finest grids.     

A higher-order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions

We propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin-boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin-boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature-based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first-order in time and second-order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.     

Local and parallel finite element algorithm for stationary incompressible magnetohydrodynamics

This article presents a local and parallel finite element method for the stationary incompressible magnetohydrodynamics problem. The key idea of this algorithm comes from the two‐grid discretization technique. Specifically, we solve the nonlinear system on a global coarse mesh, and then solve a series of linear problems on several subdomains in parallel. Furthermore, local a priori estimates are obtained on a general shape regular grid. The efficiency of the algorithm is also illustrated by some numerical experiments.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1513–1539, 2017     

Strong-stability-preserving 3-stage Hermite-Birkhoff time-discretization methods

Strong-stability-preserving (SSP) time-discretization methods have a nonlinear stability property that makes them particularly suitable for the integration of hyperbolic conservation laws. A collection of SSP explicit 3-stage Hermite-Birkhoff methods of orders 3 to 7 with nonnegative coefficients are constructed as k-step analogues of third-order Runge-Kutta methods, incorporating a function evaluation at two off-step points. Generally, these new methods have larger effective CFL coefficients than the hybrid methods of Huang with the same step number k. They have larger maximum scaled step sizes than hybrid methods on Burgers' equations.     

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

In this paper, we study the multiscale finite element discretizations about the biharmonic eigenvalue problem of plate buckling. On the basis of the work of Dai and Zhou (SIAM J. Numer. Anal. 46[1] [2008] 295‐324), we establish a three‐scale scheme, a multiscale discretization scheme, and the associated parallel version based on local defect correction. We first prove a local priori error estimate of finite element approximations, then give the error estimates of multiscale discretization schemes. Theoretical analysis and numerical experiments indicate that our schemes are suitable and efficient for eigenfunctions with local low smoothness.     

Complex dynamics of a discrete fractional‐order Leslie‐Gower predator‐prey model

A proposed discretized form of fractional‐order prey‐predator model is investigated. A sufficient condition for the solution of the discrete system to exist and to be unique is determined. Jury stability test is applied for studying stability of equilibrium points of the discretized system. Then, the effects of varying fractional order and other parameters of the systems on its dynamics are examined. The system undergoes Neimark‐Sacker and flip bifurcation under certain conditions. We observe that the model exhibits chaotic dynamics following stable states as the memory parameter α decreases and step size h increases. Theoretical results illustrate the rich dynamics and complexity of the model. Numerical simulation validates theoretical results and demonstrates the presence of rich dynamical behaviors include S‐asymptotically bounded periodic orbits, quasi‐periodicity, and chaos. The system exhibits a wide range of dynamical behaviors for fractional‐order α key parameter.     

A Note on Parallel Preconditioning for the All-at-Once Solution of Riesz Fractional Diffusion Equations

The $p$-step backward difference formula (BDF) for solving systems of ODEs can be formulated as all-at-once linear systems that are solved by parallel-in-time preconditioned Krylov subspace solvers (see McDonald et al. [36] and Lin and Ng [32]). However, when the BDF$p$ (2 ≤ $p$ ≤ 6) method is used to solve time-dependent PDEs, the generalization of these studies is not straightforward as $p$-step BDF is not selfstarting for $p$ ≥ 2. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, and show that it results into an all-at-once discretized system that is a low-rank perturbation of a block triangular Toeplitz system. We first give an estimation of the condition number of the all-at-once systems and then, capitalizing on previous work, we propose two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. An efficient implementation of these BC preconditioners is also presented, including the fast computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.     

Euler's problem,Euler's method,and the standard map; or,the discrete charm of buckling

Summary We explore the relation between the classical continuum model of Euler buckling and an iterated mapping which is not only a mathematical discretization of the former but also has an exact, discrete mechanical analogue. We show that the latter possesses great numbers of “parasitic” solutions in addition to the natural discretizations of classical buckling modes. We investigate this rich bifurcational structure using both mechanical analysis of the boundary value problem and dynamical studies of the initial value problem, which is the familiar standard map. We use this example to explore the links between discrete initial and boundary value problems and, more generally, to illustrate the complex relations among physical systems, continuum and discrete models and the analytical and numerical methods for their study.     

Stencils with isotropic discretization error for differential operators

We derive stencils, i.e., difference schemes, for differential operators for which the discretization error becomes isotropic in the lowest order. We treat the Laplacian, Bilaplacian (= biharmonic operator), and the gradient of the Laplacian both in two and three dimensions. For three dimensions ??(h²) results are given while for two dimensions both ??(h²) and ??(h⁴) results are presented. The results are also available in electronic form as a Mathematica file. It is shown that the extra computational cost of an isotropic stencil usually is less than 20%. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

基于空间离散的最短路径求解法及其局部优化方法

提出了一种基于空间离散的最短路径求解法，该法利用复杂表面的空间离散信息，从已知的两点中估算与其相连的一点的距离，递推式求取一点与其他点之间的最短距离。计算获得了各点与起点和终点的距离后，再把它们相加，依据与起点的距离的大小，顺序把距离和最小的结点连接起来，这样获得了最短路径的邻域路径，然后对最短路径的邻域路径的各点进行迭代式更新，从而获得局部优化，最终获得最短路径。经过对例子的计算及分析，表明该方法普适性强、可靠及有效。