Compact difference scheme for two-dimensional fourth-order nonlinear hyperbolic equation

High-order compact finite difference method for solving the two-dimensional fourth-order nonlinear hyperbolic equation is considered in this article. In order to design an implicit compact finite difference scheme, the fourth-order equation is written as a system of two second-order equations by introducing the second-order spatial derivative as a new variable. The second-order spatial derivatives are approximated by the compact finite difference operators to obtain a fourth-order convergence. As well as, the second-order time derivative is approximated by the central difference method. Then, existence and uniqueness of numerical solution is given. The stability and convergence of the compact finite difference scheme are proved by the energy method. Numerical results are provided to verify the accuracy and efficiency of this scheme.     

Positive numerical solution for a nonarbitrage liquidity model using nonstandard finite difference schemes

In this article, we construct a numerical method based on a nonstandard finite difference scheme to solve numerically a nonarbitrage liquidity model with observable parameters for derivatives. This nonlinear model considers that the parameters involved are observable from order book data. The proposed numerical method use a exact difference scheme in the linear convection‐reaction term, and the spatial derivative is approximated using a nonstandard finite difference scheme. It is shown that the proposed numerical scheme preserves the positivity as well as stability and consistence. To illustrate the accuracy of the method, the numerical results are compared with those produced by other methods. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 210‐221, 2014     

A Regularized Iterative Scheme for Solving Nonlinear Ill-Posed Problems

In this article, we consider a regularized iterative scheme for solving nonlinear ill-posed problems. The convergence analysis and error estimates are derived by choosing the regularization parameter according to both a priori and a posteriori methods. The iterative scheme is stopped using an a posteriori stopping rule, and we prove that the scheme converges to the solution of the well-known Lavrentiev scheme. The salient features of the proposed scheme are: (i) convergence and error estimate analysis require only weaker assumptions compared to standard assumptions followed in literature, and (ii) consideration of an adaptive a posteriori stopping rule and a parameter choice strategy that gives the same convergence rate as that of an a priori method without using the smallness assumption, the source condition. The above features are very useful from theory and application points of view. We also supply the numerical results to illustrate that the method is adaptable. Further, we compare the numerical result of the proposed method with the standard approach to demonstrate that our scheme is stable and achieves good computational output.     

Upwind compact finite difference scheme for time-accurate solution of the incompressible Navier-Stokes equations

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.     

Effect of Flux Evaluation Methods on the Resolution and Robustness of the Two-Step Finite-Difference WENO Scheme

The resolution and the robustness of the weighted essentially non-oscillatory (WENO) scheme and two-step finite-difference WENO (TSFDWENO) schemes are compared by strictly using the same flux evaluation method and smoothness indicators. TSFDWENO schemes are defined to include a family of weighted compact nonlinear scheme (WCNS) and an alternative WENO scheme. Comparison results indicate that WCNS has a higher resolution than the WENO scheme, while the WENO scheme is more robust than WCNS. Additionally, various flux evaluation methods are combined with TSFDWENO schemes, and they are evaluated. Then, the effects of the flux evaluation methods on the resolution and robustness of the scheme are investigated, and the results show that the robustness and the resolution can be significantly altered by changing the flux evaluation method. This study reveals the advantage of being able to use various flux evaluation methods in the TSFDWENO scheme as well as the fair comparison of the WENO schemes and WCNS. On the other hand, these effects are marginalized when changing the interpolation and differencing method. Such knowledge can be important when selecting schemes for actual simulation and developing guidelines for scheme improvement.     

Numerical simulation and convergence analysis for a system of nonlinear singularly perturbed differential equations arising in population dynamics

In this article, we consider a system of nonlinear singularly perturbed differential equations with two different parameters. To solve this system, we develop a weighted monotone hybrid scheme on a nonuniform mesh. The proposed scheme is a combination of the midpoint scheme and the upwind scheme involving the weight parameters. The weight parameters enable the method to switch automatically from the midpoint scheme to the upwind scheme as the nodal points start moving from the inner region to the outer region. The nonuniform mesh in particular the adaptive grid is constructed using the idea of equidistributing a positive monitor function involving the solution gradient. The method is shown to be second order convergent with respect to the small parameters. Numerical experiments are presented to show the robustness of the proposed scheme and indicate that the estimate is optimal.     

Practical acceleration for computing the HITS ExpertRank vectors

A meaningful rank as well as efficient methods for computing such a rank are necessary in many areas of applications. Major methodologies for ranking often exploit principal eigenvectors. Kleinberg’s HITS model is one of such methodologies. The standard approach for computing the HITS rank is the power method. Unlike the PageRank calculations where many acceleration schemes have been proposed, relatively few works on accelerating HITS rank calculation exist. This is mainly because the power method often works quite well in the HITS setting. However, there are cases where the power method is ineffective, moreover, a systematic acceleration over the power method is desirable even when the power method works well. We propose a practical acceleration scheme for HITS rank calculations based on the filtered power method by adaptive Chebyshev polynomials. For cases where the gap-ratio is below 0.85 for which the power method works well, our scheme is about twice faster than the power method. For cases where gap-ratio is unfavorable for the power method, our scheme can provide significant speedup. When the ranking problems are of very large scale, even a single matrix–vector product can be expensive, for which accelerations are highly necessary. The scheme we propose is desirable in that it provides consistent reduction in number of matrix–vector products as well as CPU time over the power method, with little memory overhead.     

A simple adaptive-feedback controller for chaos synchronization

Based on the LaSalle’s invariant theorem and Lyapunov method, a simple scheme is proposed to synchronize chaotic systems. Unlike the usual linear feedback, this scheme uses the variable feedback which is automatically adapted to a updated law. Moreover, this scheme is analytical and simple to be implemented in practice. The well-known models such as Chen system and Lü system are used to illustrate the validity of this theoretic method.     

Block preconditioning strategies for nonlinear viscous wave equations

In this paper, five block preconditioning strategies are proposed to solve a class of nonlinear viscous wave equations. Implicit time-integration techniques from low order to high order are considered exclusively including implicit Euler (IE1) method, backward differentiation formulas (BDF2, BDF3) as well as the Crank–Nicholson (CN2) scheme. The CN2 method demonstrates superior performance compared to the BDF2 scheme for the problems considered in this work. In addition, the third-order accurate BDF3 scheme is found to be the most efficient in terms of computational cost for a prescribed accuracy level. Moreover, the benefit of this scheme increases for tighter error tolerances.     

A Newton linearized compact finite difference scheme for one class of Sobolev equations

In this article, a Newton linearized compact finite difference scheme is proposed to numerically solve a class of Sobolev equations. The unique solvability, convergence, and stability of the proposed scheme are proved. It is shown that the proposed method is of order 2 in temporal direction and order 4 in spatial direction. Moreover, compare to the classical extrapolated Crank‐Nicolson method or the second‐order multistep implicit–explicit methods, the proposed scheme is easier to be implemented as it only requires one starting value. Finally, numerical experiments on one and two‐dimensional problems are presented to illustrate our theoretical results.     

Left distance binary tree representations

We present a new scheme for representing binary trees. The scheme is based on rotations as a previous scheme of Zerling. In our method the items of a representation have a natural geometric interpretation, and the algorithms related to the method are simple. We give an algorithm for enumerating all the representations for trees onn nodes, and an algorithm for building the tree corresponding to a given representation.This work was supported by the Academy of Finland.     

Two-phase laminar axisymmetric jet flow: Explicit,implicit, and split-operator approximations

An hybrid Eulerian-Lagrangian numerical scheme is developed for a two-phase problem and four finite-difference schemes are compared. For this purpose, the problem of hydrodynamic and thermal interactions between a fuel spray and a mixing region of two laminar, unconfined axisymmetric jets is formulated in terms of a set of parabolic differential equations for the gas phase and a set of Lagrangian ordinary differential equations for the condensed phase. Consistent, second-order accurate hybrid numerical schemes, with the exception of the explicit scheme with an accuracy between linear and quadratic, are used to solve these equations. The subset of gas-phase equations has been solved by four different numerical methods: a predictor-corrector explicit method, a sequential implicit method, a block implicit method, and a symmetric operator-splitting method. The subsystem of liquid-phase equations is solved along the droplet trajectories by a second-order Runge-Kutta scheme. The computations have been made to predict the hydro-dynamic and thermal mixing regions of the gas phase as well as the trajectories of each individual group of droplets. In addition, the size, velocity and temperature associated with each group are predicted along these trajectories. The relative merits of the above four difference-schemes are discussed by constructing effectiveness curves. At low error tolerances, the sequential implicit method gives the best results, where for large error tolerances, the explicit and operator splitting give better results. The block implicit scheme is the least effective at all accuracy requirements.     

Analysis of Hermite subdivision using piecewise polynomials

In this paper we begin to explore a new method of analyzing the regularity of Hermite subdivision schemes that are defined from local polynomial interpolants. The idea of the method is to view the limit of the scheme as the limit of splines formed by these local interpolants rather than as the limit of polygons. We demonstrate the success of the method by obtaining the precise Hölder regularity of the simple, but non-trivial scheme in which the data are uniformly spaced and the refinement rule is defined by quintic interpolation of four values and two derivatives.     

An Explicit Pseudo-Spectral Scheme with Almost Unconditional Stability for the Cahn-Hilliard Equation

In this paper, an explicit fully discrete three-level pseudo-spectral scheme with almost unconditional stability for the Cahn-Hilliard equation is proposed. Stability and convergence of the scheme are proved by Sobolev's inequalities and the bounded extensive method of the nonlinear function (B.N. Lu (1995)). The scheme possesses the almost same stable condition and convergent accuracy as the Creak-Nicloson scheme but it is an explicit scheme. Thus the iterative method to solve nonlinear algebraic system is avoided. Moreover, the linear stability of the critical point $u_0$ is investigated and the linear dispersive relation is obtained. Finally, the numerical results are supplied, which check the theoretical results.     

How to increase convergence order of the Newton method to 2 × m?

We present a simple and effective scheme for forming iterative methods of various convergence orders. In this scheme, methods of various convergence orders, such as four, six, eight and ten, are formed through a modest modification of the classical Newton method. Since the scheme considered is a simple modification of the Newton method, it can be easily implemented in existing software packages, which is also suggested by the presented pseudocodes. Finally some problems are solved, to very high precision, through the proposed scheme. Numerical work suggests that the presented scheme requires less number of function evaluations for convergence and it may be suitable in high precision computing.     

On the numerical solution of the Klein‐Gordon equation

A predictor–corrector (P–C) scheme based on the use of rational approximants of second‐order to the matrix‐exponential term in a three‐time level reccurence relation is applied to the nonlinear Klein‐Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double‐soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A composite Chebyshev finite difference method for nonlinear optimal control problems

In this paper, a composite Chebyshev finite difference method is introduced and is successfully employed for solving nonlinear optimal control problems. The proposed method is an extension of the Chebyshev finite difference scheme. This method can be regarded as a non-uniform finite difference scheme and is based on a hybrid of block-pulse functions and Chebyshev polynomials using the well-known Chebyshev–Gauss–Lobatto points. The convergence of the method is established. The nice properties of hybrid functions are then used to convert the nonlinear optimal control problem into a nonlinear mathematical programming one that can be solved efficiently by a globally convergent algorithm. The validity and applicability of the proposed method are demonstrated through some numerical examples. The method is simple, easy to implement and yields very accurate results.     

Derivative-Free Filter Simulated Annealing Method for Constrained Continuous Global Optimization

In this paper, a simulated-annealing-based method called Filter Simulated Annealing (FSA) method is proposed to deal with the constrained global optimization problem. The considered problem is reformulated so as to take the form of optimizing two functions, the objective function and the constraint violation function. Then, the FSA method is applied to solve the reformulated problem. The FSA method invokes a multi-start diversification scheme in order to achieve an efficient exploration process. To deal with the considered problem, a filter-set-based procedure is built in the FSA structure. Finally, an intensification scheme is applied as a final stage of the proposed method in order to overcome the slow convergence of SA-based methods. The computational results obtained by the FSA method are promising and show a superior performance of the proposed method, which is a point-to-point method, against population-based methods.     

A roe-type Riemann solver for hyperbolic systems with relaxation based on time-dependent wave decomposition

Summary. The paper is devoted to the construction of a higher order Roe-type numerical scheme for the solution of hyperbolic systems with relaxation source terms. It is important for applications that the numerical scheme handles both stiff and non stiff source terms with the same accuracy and computational cost and that the relaxation variables are computed accurately in the stiff case. The method is based on the solution of a Riemann problem for a linear system with constant coefficients: a study of the behavior of the solutions of both the nonlinear and linearized problems as the relaxation time tends to zero enables to choose a convenient linearization such that the numerical scheme is consistent with both the hyperbolic system when the source terms are absent and the correct relaxation system when the relaxation time tends to zero. The method is applied to the study of the propagation of sound waves in a two-phase medium. The comparison between our numerical scheme, usual fractional step methods, and numerical simulation of the relaxation system shows the necessity of using the solutions of a fully coupled hyperbolic system with relaxation terms as the basis of a numerical scheme to obtain accurate solutions regardless of the stiffness. Received October 7, 1994 / Revised version received September 27, 1995     

From the fitting techniques to accurate schemes for the Liouville‐Bratu‐Gelfand problem

We are interested in numerical methods for the Liouville‐Bratu‐Gelfand problem. The ideas and techniques developed here to construct the schemes are inspired from the fitted method and the so‐called compact exponentially fitted method. Some of those schemes can be viewed as extensions of both the Buckmire scheme and the standard scheme which results from the use of the standard finite‐difference procedures. We study and compare computationally the accuracy of methods introduced here. It is also mentioned that the Buckmire's techniques and the standard scheme are a particular case of the fitted method. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006