Solving delay differential equations by the five-point one-step block method using Neville's interpolation

A five-point one-step block method based on the Newton backward divided difference formulae for the solution of first-order delay differential equations is derived. The proposed block method will approximate the solutions of initial value problems at five points simultaneously using variable step size. The approximation of the delay term is calculated using Neville's interpolation. The block method will be formulated in terms of linear multistep method, but the method is equivalent to one-step method. The order of the block method will be discussed. The P-stability and Q-stability regions of the block method using Neville's interpolation for the delay term are presented for a fixed step size. Numerical results are given to show the efficiency of the proposed method and compared with the existing method.     

Numerical simulation of trajectory prescribed path control problems by the backward differentiation formulas

The equations which describe trajectory, prescribed path control problems naturally form nonlinear semiexplicit, differential-algebraic systems with index greater than one. It is known that not all fully implicit systems may be solved stably by the k-step backward differentiation formulas, yet these methods do produce convergent numerical solutions to some semiexplicit systems. In this note numerical results are presented for the simplest backward differentiation formula when applied to an index three, semiexplicit system concerning the reentry of the space shuttle. The application of this numerical method to a realistic problem illustrates some unresolved implementation difficulties.     

An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation

In this paper, we present an unconditional stable linear high-order finite difference scheme for three dimensional Allen–Cahn equation. This scheme, which is based on a backward differentiation scheme combined with a fourth-order compact finite difference formula, is second order accurate in time and fourth order accurate in space. A linearly stabilized splitting scheme is used to remove the restriction of time step. We prove the unconditional stability of our proposed method in analysis. A fast and efficient linear multigrid solver is employed to solve the resulting discrete system. We perform various numerical experiments to confirm the high-order accuracy, unconditional stability and efficiency of our proposed method. In particular, we show two applications of our proposed method: triply-periodic minimal surface and volume inpainting.     

Implementation of parallel three-point block codes for solving large systems of ordinary differential equations

The three-point fully implicit block methods are developed for solving large systems of ordinary differential equations using variable step size on a parallel shared memory computer. The methods calculate the numerical solution at three points simultaneously and are suitable for parallelization across the method. The methods are in a simple form as Adams Moulton method with the specific aim of gaining efficiency. For large problems, the parallel implementation produced a good speed-up with respect to the sequential timing and hence better efficiency for the methods developed.     

Error analysis of an algorithm for equality-constrained quadratic programming problems

In this paper the numerical stability of the orthogonal factorization method [5] for linear equality-constrained quadratic programming problems is studied using a backward error analysis. A perturbation formula for the problem is analyzed; the condition numbers of this formula are examined in order to compare them with the condition numbers of the two matrices of the problem. A class of test problems is also considered in order to show experimentally the behaviour of the method.     

Tseng type methods for solving inclusion problems and its applications

In this paper, we introduce two modifications of the forward–backward splitting method with a new step size rule for inclusion problems in real Hilbert spaces. The modifications are based on Mann and viscosity-ideas. Under standard assumptions, such as Lipschitz continuity and monotonicity (also maximal monotonicity), we establish strong convergence of the proposed algorithms. We present two numerical examples, the first in infinite dimensional spaces, which illustrates mainly the strong convergence property of the algorithm. For the second example, we illustrate the performances of our scheme, compared with the classical forward–backward splitting method for the problem of recovering a sparse noisy signal. Our result extend some related works in the literature and the primary experiments might also suggest their potential applicability.     

Numerical simulation of contaminant biodegradation by higher order methods and adaptive time stepping

In this work we present and analyze a reliable and robust approximation scheme for biochemically reacting transport in the subsurface following Monod type kinetics. Water flow is modeled by the Richards equation. The proposed scheme is based on higher order finite element methods for the spatial discretization and the two step backward differentiation formula for the temporal one. The resulting nonlinear algebraic systems of equations are solved by a damped version of Newtons method. For the linear problems of the Newton iteration Krylov space methods are used. In computational experiments conducted for realistic subsurface (groundwater) contamination scenarios we show that the higher order approximation scheme significantly reduces the amount of inherent numerical diffusion compared to lower order ones. Thereby an artificial transverse mixing of the species leading to a strong overestimation of the biodegradation process is avoided. Finally, we present a robust adaptive time stepping technique for the coupled flow and transport problem which allows efficient long-term predictions of biodegradation processes.     

基于压缩感知的后退型自适应匹配追踪算法

基于压缩感知理论的重建关键在于从压缩感知得到的低维数据中精确恢复出原始的高维稀疏数据。针对目前大多数算法都建立在稀疏度已知的基础上,提出一种后退型固定步长自适应匹配追踪重建算法,能够在稀疏度未知的条件下获得图像的精确重建。该算法通过较大固定步长的设置,保证待估信号支撑集大小的稳步快速增加;以相邻阶段重建信号的能量差为迭代停止条件,在迭代停止后通过简单的正则化方法向后剔除多余原子保证精确重建。实验结果表明,该算法在保证测量次数的条件下可以获得快速的精确重建。     

A finite volume–finite difference method with a stiff ordinary differential equation solver for advection–diffusion–reaction equation

A model for atmospheric pollutant transport is proposed considering an advection–diffusion–reaction equation. A splitting method is used to decouple the advection, diffusion and reaction parts. A scheme based on finite volume, finite difference and backward differentiation formula is used for solving an atmospheric transport-chemistry problem.     

A high-order HDG method for the Biot’s consolidation model

We propose a novel high-order HDG method for the Biot’s consolidation model in poroelasticity. We present optimal $h$-version error analysis for both the semi-discrete and full-discrete (combined with temporal backward differentiation formula) schemes. Numerical tests are provided to demonstrate the performance of the method.     

A(α)-stable cyclic composite multistep methods of order 5

We describe a newA(α)-stable 3-cyclic 3-step method with accuracy of orderP=5. This is in contrast to the originally developed methods [5] which have greater order of accuracy (P=6) but a small stability region. Moreover, the new method shows slightly better stability than the backward differentiation formula (BDF) [6] of the same order.     

A collocation method based on the influence matrix technique for Navier-Stokes problems in annular domains

A spectral collocation method is proposed for the solution of the time-dependent Navier-Stokes and energy equations of a Boussinesq fluid inside an annular cavity. The time integration is based on the Adams-Bashforth scheme and on the second order backward differentiation formula. The influence matrix technique results in the resolution of Helmholtz and Poisson equations with Dirichlet boundary conditions. The solutions are validated with respect to former spectral Tau-Chebyshev solutions. Preliminary results concern the simulation of axisymmetric flows submitted to the buoyancy force, to the rotation and to source-sink fluxes.     

细菌觅食算法求解高维优化问题

针对细菌觅食优化算法中,以往的自适应步长公式引入参数过多,统一的经验性参数无法适应各类不同问题的情况,提出了改进的自适应步长公式,通过在步长公式中引入当前细菌的进化代数、寻优范围,并发挥当前最优细菌的引导作用,灵活的调整步长,真正达到自适应调整步长的目的;其次对高维优化问题进行分析,将其分为可分解可分组、不可分解可分组和不可分解不可分组三大类,针对不同类型的问题,采用不同的分组方式,降维、细化来求解,将复杂的问题简单化,极大的提高了求解的效率和精度。将改进的自适应步长公式应用于高维优化问题的求解方法中,通过对多个标准测试函数在多维空间特别是超高维空间（500维、800维、1000维）进行测试,并将其结果同其它算法进行比较,实验证明本文改进算法在寻得最优解的精度和效率上比其它改进方案有显著提高。     

Stabilized discontinuous Galerkin method for hyperbolic equations

In this work a new stabilization technique is proposed and studied for the discontinuous Galerkin method applied to hyperbolic equations. In order to avoid the use of slope limiters, a streamline diffusion-like term is added to control oscillations for arbitrary element orders. Thus, the scheme combines ideas from both the Runge-Kutta discontinuous Galerkin method [J. Scient. Comput. 16 (2001) 173] and the streamline diffusion method [Comput. Methods Appl. Mech. Engrg. 32 (1982)]. To increase the stability range of the method, the diffusion term is treated implicitly. The result is a scheme with higher order in space with the same stability range as the finite volume method. An optimal relation between the time step and the size of the diffusion coefficient is analyzed for numerical precision. The scheme is implemented using the object oriented programming philosophy based on the environment described in [Comput. Methods Appl. Mech. Engrg. 150 (1997)]. Accuracy and shock capturing abilities of the method are analyzed in terms of two bidimensional model problems: the rotating cone and the backward facing step problem for the Euler equations of gas dynamics.     

Fast leave-one-out evaluation for dynamic gene selection

Gene selection procedure is a necessary step to increase the accuracy of machine learning algorithms that help in disease diagnosis based on gene expression data. This is commonly known as a feature subset selection problem in machine learning domain. A fast leave-one-out (LOO) evaluation formula for least-squares support vector machines (LS-SVMs) is introduced here that can guide our backward feature selection process. Based on that, we propose a fast LOO guided feature selection (LGFS) algorithm. The gene selection step size is dynamically adjusted according to the LOO accuracy estimation. For our experiments, the application of LGFS to the gene selection process improves the classifier accuracy and reduces the number of features required as well. The least number of genes that can maximize the disease classification accuracy is automatically determined by our algorithm.     

Quasi-wavelet method for time-dependent fractional partial differential equation

In this decade, many new applications in engineering and science are governed by a series of fractional partial differential equations. In this paper, we propose a novel numerical method for a class of time-dependent fractional partial differential equations. The time variable is discretized by using the second order backward differentiation formula scheme, and the quasi-wavelet method is used for spatial discretization. The stability and convergence properties related to the time discretization are discussed and theoretically proven. Numerical examples are obtained to investigate the accuracy and efficiency of the proposed method. The comparisons of the present numerical results with the exact analytical solutions show that the quasi-wavelet method has distinctive local property and can achieve accurate results.     

Error estimation and control for ODEs

This article is about the numerical solution of initial value problems for systems of ordinary differential equations. At first these problems were solved with a fixed method and constant step size, but nowadays the general-purpose codes vary the step size, and possibly the method, as the integration proceeds. Estimating and controlling some measure of error by variation of step size/method inspires some confidence in the numerical solution and makes possible the solution of hard problems. Common ways of doing this are explained briefly in the article.     

Error Estimation and Control for ODEs

This article is about the numerical solution of initial value problems for systems of ordinary differential equations. At first these problems were solved with a fixed method and constant step size, but nowadays the general-purpose codes vary the step size, and possibly the method, as the integration proceeds. Estimating and controlling some measure of error by variation of step size/method inspires some confidence in the numerical solution and makes possible the solution of hard problems. Common ways of doing this are explained briefly in the article     

Chaos induced by the generalized Euler method

In this paper, bifurcation and the chaotic phenomenon in numerical computation are investigated using the generalized Euler method. In this method, the forward and backward Euler methods and the trapezoidal method can be represented as special cases. It will be made clear that, for some computation step size, the solution of the difference equation does not always approximate to the solution of the original differential equation; period-doubling bifurcations and the chaotic phenomenon occur in the solution of the difference equation. Two examples are shown to illustrate the effect of step size.     

Fourth-order runge-kutta schemes for fluid mechanics applications

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier-Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit-Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection-diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge-Kutta (ARK₂) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK₂ schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge-Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.