Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix

This paper presents the generalized nonlinear delay differential equations of fractional variable-order. In this article, a novel shifted Jacobi operational matrix technique is introduced for solving a class of multi-terms variable-order fractional delay differential equations via reducing the main problem to an algebraic system of equations that can be solved numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical experiments are presented to demonstrate the efficiency, generality, accuracy of proposed scheme and the flexibility of this method. The numerical results compared it with other existing methods such as fractional Adams method (FAM), new predictor–corrector method (NPCM), a new approach, Adams–Bashforth–Moulton algorithm and L1 predictor–corrector method (L1-PCM). Comparing the results of these methods as well as comparing the current method (NSJOM) with the exact solution, indicating the efficiency and validity of this method. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated.

     

A modified explicit numerical scheme for the two-dimensional sine-Gordon equation

Abstract

A fourth-order rational approximant to the matrix-exponential term in a three-time-level recurrence relation is used to transform the two-dimensional sine-Gordon equation into a second-order initial-value problem. The resulting nonlinear system is solved using an appropriate predictor–corrector (P-C) scheme in which the predictor is an explicit one of second order. The procedure of the corrector is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the nonlinear method and the predictor–corrector are analysed for local truncation error and stability. The MPC scheme has been tested on line and circular ring solitons known from the literature, and numerical experiments have proved that there is an improvement in accuracy over the standard predictor–corrector implementation.     

Root locus for SISO dead-time systems: A continuation based approach

We present a numerical method to plot the root locus of Single-Input–Single-Output (SISO) dead-time systems with respect to the controller gain or the system delay. We compute the trajectories of characteristic roots of the closed-loop system on a prescribed complex right half-plane. We calculate the starting, branch and boundary crossing roots of root-locus branches inside the region. We compute the root locus of each characteristic root based on a predictor–corrector type continuation method. To avoid the high sensitivity of roots with respect to the locus parameter in the neighborhood of branch points, the continuation method relies on a natural parameterization of the root-locus trajectory in terms of a distance in the (characteristic root, locus parameter)-space. The method is numerically stable for high order SISO dead-time systems.     

A fourth order numerical scheme for the one-dimensional sine-Gordon equation

A numerical scheme arising from the use of a fourth order rational approximants to the matrix-exponential term in a three-time level recurrence relation is proposed for the numerical solution of the one-dimensional sine-Gordon (SG) equation already known from the bibliography. The method for its implementation uses a predictor–corrector scheme in which the corrector is accelerated by using the already evaluated corrected values modified predictor–corrector scheme. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. Both the predictor and the corrector schemes are analysed for stability. The predictor–corrector/modified predictor–corrector (P-C/MPC) schemes are tested on single and soliton doublets as well as on the collision of breathers and a comparison of the numerical results with the corresponding ones in the bibliography is made. Finally, conclusions for the behaviour of the introduced MPC over the standard P-C scheme are derived.     

Euler’s predictor–corrector technique for solving the depth-integrated shallow water equations

Euler’s predictor–corrector technique combined with finite analysis method is applied to solve 2D advection–diffusion shallow water equations. In this algorithm the momentum equations are calculated by the finite analysis method based on a single mesh, while the continuity equation is solved by Euler’s predictor–corrector technique. To verify the performance of this approach, the simulation of tidal flow in the Huangpu estuary is carried out. The numerical results are found to be consistent with the field results, implying that this proposed method is effective and applicable.     

A Positivity-Preserving Well-Balanced Central Discontinuous Galerkin Method for the Nonlinear Shallow Water Equations

In this paper, we consider the development of central discontinuous Galerkin methods for solving the nonlinear shallow water equations over variable bottom topography in one and two dimensions. A reliable numerical scheme for these equations should preserve still-water stationary solutions and maintain the non-negativity of the water depth. We propose a high-order technique which exactly balances the flux gradients and source terms in the still-water stationary case by adding correction terms to the base scheme, meanwhile ensures the non-negativity of the water depth by using special approximations to the bottom together with a positivity-preserving limiter. Numerical tests are presented to illustrate the accuracy and validity of the proposed schemes.     

A stable Gaussian radial basis function method for solving nonlinear unsteady convection–diffusion–reaction equations

We investigate a novel method for the numerical solution of two-dimensional time-dependent convection–diffusion–reaction equations with nonhomogeneous boundary conditions. We first approximate the equation in space by a stable Gaussian radial basis function (RBF) method and obtain a matrix system of ODEs. The advantage of our method is that, by avoiding Kronecker products, this system can be solved using one of the standard methods for ODEs. For the linear case, we show that the matrix system of ODEs becomes a Sylvester-type equation, and for the nonlinear case we solve it using predictor–corrector schemes such as Adams–Bashforth and implicit–explicit (IMEX) methods. This work is based on the idea proposed in our previous paper (2016), in which we enhanced the expansion approach based on Hermite polynomials for evaluating Gaussian radial basis function interpolants. In the present paper the eigenfunction expansions are rebuilt based on Chebyshev polynomials which are more suitable in numerical computations. The accuracy, robustness and computational efficiency of the method are presented by numerically solving several problems.     

A Numerical Scheme for a Viscous Shallow Water Model with Friction

We consider a particular viscous shallow water model with topography and friction laws, formally derived by asymptotic expansion from the three-dimensional free surface Navier-Stokes equations. Emphasize is put on the numerical study: the viscous system is regarded as an hyperbolic system with source terms and discretized using a second order finite volume method. New steady states solutions for open channel flows are introduced for the whole model with viscous and friction terms. The proposed numerical scheme is validated against these new benchmarks.     

Extension of WAF Type Methods to Non-Homogeneous Shallow Water Equations with Pollutant

This paper deals with the extension of the WAF method to discretize Shallow Water Equations with pollutants. We consider two different versions of the WAF method, by approximating the intermediate waves using the flux of HLL or the direct approach of HLLC solver. It is seen that both versions can be written under the same form with different definitions for the approximation of the velocity waves. We also propose an extension of the method to non-homogeneous systems. In the case of homogeneous systems it is seen that we can rewrite the third component of the numerical flux in terms of an intermediate wave speed approximation. We conclude that—in order to have the same relation for non-homogeneous systems—the approximation of the intermediate wave speed must be modified. The proposed extension of the WAF method preserves all stationary solutions, up to second order accuracy, and water at rest in an exact way, even with arbitrary pollutant concentration. Finally, we perform several numerical tests, by comparing it with HLLC solver, reference solutions and analytical solutions.     

Enriched Spectral Method for Stiff Convection-Dominated Equations

A novel and simple numerical method for stiff convection-dominated problems is studied in presence of boundary or interior layers. A version of the spectral Chevyshev-collocation method enriched with the so-called corrector functions is investigated. The corrector functions here are designed to capture the stiffness of the layers (see the Appendix), and the proposed method does not rely on the adaptive grid points. The extensive numerical results demonstrate that the enriched spectral methods are very accurate with low computational cost.     

Predictor–corrector Obreshkov pairs

The combination of predictor–corrector (PEC) pairs of Adams methods can be generalized to high derivative methods using Obreshkov quadrature formulae. It is convenient to construct predictor–corrector pairs using a combination of explicit (Adams–Bashforth for traditional PEC methods) and implicit (Adams–Moulton for traditional PEC methods) forms of the methods. This paper will focus on one special case of a fourth order method consisting of a two-step predictor followed by a one-step corrector, each using second derivative formulae. There is always a choice in predictor–corrector pairs of the so-called mode of the method and we will consider both PEC and PECE modes. The Nordsieck representation of Adams methods, as developed by C. W. Gear and others, adapts well to the multiderivative situation and will be used to make variable stepsize convenient. In the first part of the paper we explain the basic approximations used in the predictor–corrector formula. Those can be written in terms of Obreshkov quadrature. Next section we discuss the equations in terms of Nordsieck vectors. This provides an opportunity to extend the Gear Nordsieck factorization to achieve a variable stepsize formulation. Numerical tests with the new method are also discussed. The paper will present Prothero–Robinson and Kepler problem to illustrate the power of the approach.     

A low Mach number scheme based on multi-scale asymptotics

The low Mach number regime is characterized by a large discrepancy between the flow velocity and the speed of sound, leading to physical effects on different length scales and of different orders of magnitude. A single time scale, multiple space scale asymptotic analysis provides detailed insight into the limit behavior of solutions of the compressible Euler equations as the Mach number tends to zero. This analysis shows that “the pressure” splits up into three parts with different physical meanings. This knowledge is then used to develop a numerical scheme including multiple pressure variables to account for the different effects. The numerical method is a semi-implicit predictor-corrector algorithm. In the predictor step, the asymptotic equations are used to guess the global and large scale effects. Then the corrector step can be viewed as an incompressible solver with compressibility effects acting as source terms. Received: 28 July 1999 / Accepted: 21 September 1999     

混联复合机床转台定位精度的激光测量与补偿

传统的数控机床进给轴定位精度的检测方法,存在精度低、方法落后、检验重复性差等缺点。因此,采用目前国际先进的激光干涉法检测混联复合机床转台的定位精度,以评价和提高其精度。对干涉仪的测量原理、测量方法和数据处理进行了分析,认为空气折射率是干涉仪应用中的主要误差来源,并分析了其原因和补偿方法。最后,给出了基于激光干涉仪的机床定位精度检测和补偿方案。     

Numerical Treatment of the Loss of Hyperbolicity of?the?Two-Layer Shallow-Water System

This article is devoted to the numerical solution of the inviscid two-layer shallow water system. This system may lose the hyperbolic character when the shear between the layer is big enough. This loss of hyperbolicity is related to the appearance of shear instabilities that leads, in real flows, to intense mixing of the two layers that the model is not able to simulate. The strategy here is to add some extra friction terms, which are supposed to parameterize the loss of mechanical energy due to mixing, to get rid of this difficulty. The main goal is to introduce a technique allowing one to add locally and automatically an ??optimal?? amount of shear stress to make the flow to remain in the hyperbolicity region. To do this, first an easy criterium to check the hyperbolicity of the system for a given state is proposed and checked. Next, we introduce a predictor/corrector strategy. In the predictor stage, a numerical scheme is applied to the system without extra friction. In the second stage, a discrete semi-implicit linear friction law is applied at any cell in which the predicted states are not in the hyperbolicity region. The coefficient of this law is calculated so that the predicted states are driven to the boundary of the hyperbolicity region according to the proposed criterium. The numerical scheme to be used at the first stage has to be able to advance in time in presence of complex eigenvalues: we propose here a family of path-conservative numerical scheme having this property. Finally, some numerical tests have been performed to assess the efficiency of the proposed strategy.     

微分方程初值问题的GDQR解法

本文用最近由Wu提出的一种数值方法-GDQR(GeneralizedDifferentialQuadra-tureRule)对工程和科学技术中常遇到的2—4阶微分方程初值问题进行了求解.部分结果与精确解或龙格-库塔方法所得结果作了对比,表明GDQR在解决常微分方程初值问题时简单方便有效.     

A Not-a-Knot meshless method using radial basis functions and predictor–corrector scheme to the numerical solution of improved Boussinesq equation

A numerical simulation of the improved Boussinesq (IBq) equation is obtained using collocation and approximating the solution by radial basis functions (RBFs) based on the third-order time discretization. To avoid solving the nonlinear system, a predictor–corrector scheme is proposed and the Not-a-Knot method is used to improve the accuracy in the boundary. The method is tested on two problems taken from the literature: propagation of a solitary wave and interaction of two solitary waves. The results of numerical experiments are compared with analytical solution and with those of other recently published methods to confirm the accuracy and efficiency of the new scheme presented in this paper.     

Lax–Friedrichs Multigrid Fast Sweeping Methods for Steady State Problems for Hyperbolic Conservation Laws

Fast sweeping methods are efficient Gauss–Seidel iterative numerical schemes originally designed for solving static Hamilton–Jacobi equations. Recently, these methods have been applied to solve hyperbolic conservation laws with source terms. In this paper, we propose Lax–Friedrichs fast sweeping multigrid methods which allow even more efficient calculations of viscosity solutions of stationary hyperbolic problems. Due to the choice of Lax–Friedrichs numerical fluxes, general problems can be solved without difficult inversion. High order discretization, e.g., WENO finite difference method, can be incorporated to achieve high order accuracy. On the other hand, multigrid methods, which have been widely used to solve elliptic equations, can speed up the computation by smoothing errors of low frequencies on coarse meshes. We modify the classical multigrid method with regard to properties of viscous solutions to hyperbolic conservation equations by introducing WENO interpolation between levels of mesh grids. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency, high order accuracy and the capability of resolving singularities of the viscosity solutions.     

Topology optimization for minimum stress design with the homogenization method

This paper is devoted to minimum stress design in structural optimization. The homogenization method is extended to such a framework and yields an efficient numerical algorithm for topology optimization. The main idea is to use a partial relaxation of the problem obtained by introducing special microstructures which are sequential laminated composites. Indeed, the so-called corrector terms of such microgeometries are explicitly known, which allows us to compute the relaxed objective function. These correctors can be interpreted as stress amplification factors, caused by the underlying microstructure.     

Optimally Convergent HDG Method for Third-Order Korteweg–de Vries Type Equations

We develop and analyze a new hybridizable discontinuous Galerkin method for solving third-order Korteweg–de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton–Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.     

基于标准白色模板的扩展RGB颜色校正算法

如何通过颜色校正获得真实的颜色重现已成为图像处理中普遍存在的技术难点。针对此类问题,提出一种基于白色模板的颜色校正方法：首先将颜色空间的精度扩展到一定的程度,然后制作一个标准状态下的白色模板;接着根据条件选取适当的校正算子,根据校正算子得出的RGB值去校正在同样条件下得到的目标图像。工程实践和仿真结果表明,校正后的图像不但在视觉上与标准图像达到一致,并且在数值误差上也小于传统的校正方法。