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Abstract: Comment on the paper “A class of methods based on non-polynomial spline functions for the solution of a special fourth-order boundary-value problems with engineering applications” in Applied Mathematics and Computation 174 (2006) 1169–1180. The paper considered a class of two-point boundary-value problems of the form where f(x) and g(x) are continuous on [a, b], and A i and B i (i =1,2) are finite real constants. Here we correct some mistake in derivation of non-polynomial spline, boundary formulas, truncation errors, convergence analysis and computational experiments. [Copyright &y& Elsevier]

Abstract Twenty years after the classical book of Ortega and Rheinboldt was published, five definitions for the Q -convergence orders of sequences were independently and rigorously studied (i.e., some orders characterized in terms of others), by Potra (1989), resp. Beyer, Ebanks and Qualls (1990). The relationship between all the five definitions (only partially analyzed in each of the two papers) was not subsequently followed and, moreover, the second paper slept from the readers attention. The main aim of this paper is to provide a rigorous, selfcontained, and, as much as possible, a comprehensive picture of the theoretical aspects of this topic, as the current literature has taken away the credit from authors who obtained important results long ago. Moreover, this paper provides rigorous support for the numerical examples recently presented in an increasing number of papers, where the authors check the convergence orders of different iterative methods for solving nonlinear (systems of) equations. Tight connections between some asymptotic quantities defined by theoretical and computational elements are shown to hold. [ABSTRACT FROM AUTHOR]

This paper presents a robust finite-time guidance (RFTG) law to a short-range interception problem. The main challenge is that the evasive strategy of the target is unpredictable because it is determined not only by the states of both the interceptor and the target, but also by external un-modeled factors. By robustly stabilizing a line-of-sight rate, this paper proposes an integrated continuous finite-time disturbance observer/bounded continuous finite-time stabilizer strategy. The design of this integrated strategy has two points: 1) effect of a target maneuver is modeled as disturbance and then is estimated by the second-order homogeneous observer; 2) the finite-time stabilizer is actively coupled with the observer. Based on homogeneity technique, the local finite-time input-to-state stability is established for the closed-loop guidance system, thus implying the proposed RFTG law can quickly render the LOS rate within a bounded error throughout intercept. Moreover, convergence properties of the LOS rate in the presence of control saturation are discussed. Numerical comparison studies demonstrate the guidance performance. [ABSTRACT FROM AUTHOR]

In this paper, a modified weak Galerkin method is proposed for the Stokes problem. The numerical scheme is based on a novel variational form of the Stokes problem. The degree of freedoms in the modified weak Galerkin method is less than that in the original weak Galerkin method, while the accuracy stays the same. In this paper, the optimal convergence orders are given and some numerical experiments are presented to verify the theory. [ABSTRACT FROM AUTHOR]

This paper is devoted to numerical global optimization algorithms applying several ideas to reduce the problem dimension. Two approaches to the dimensionality reduction are considered. The first one is based on the nested optimization scheme that reduces the multidimensional problem to a family of one-dimensional subproblems connected in a recursive way. The second approach as a reduction scheme uses Peano-type space-filling curves mapping multidimensional domains onto one-dimensional intervals. In the frameworks of both the approaches, several univariate algorithms belonging to the characteristical class of optimization techniques are used for carrying out the one-dimensional optimization. Theoretical part of the paper contains a substantiation of global convergence for the considered methods. The efficiency of the compared global search methods is evaluated experimentally on the well-known GKLS test class generator used broadly for testing global optimization algorithms. Results for representative problem sets of different dimensions demonstrate a convincing advantage of the adaptive nested optimization scheme with respect to other tested methods. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of a first order fractional-step time-scheme (via decomposition of the viscosity) and “inf-sup” stable finite-element spatial approximations applied to the Primitive Equations of the Ocean. The aim of the paper is twofold. First, we prove that the scheme is unconditionally stable and convergent towards weak solutions of the Primitive Equations. Second, optimal error estimates for velocity and pressure are provided of order O ( k + h l ) for l = 1 or l = 2 considering either first or second order finite-element approximations ( k and h being the time step and the mesh size, respectively). In both cases, these error estimates are obtained under the same constraint k ≤ C h 2 . [ABSTRACT FROM AUTHOR]

This paper discusses the continuous-time consensus problems for directed multi-agent networks under certain coupling or control. Since agents are driven not only by their neighbors' states but also by their derivatives in many realistic situations, dynamical networks described by singular systems are appropriate to study. Consensus problems for such multi-agent networks are considered when the agents communicate in the presence or absence of time delays. The maximum tolerated time-delay is obtained when the multi-agent network asymptotically reaches consensus. The results of this paper indicate that such multi-agent networks can achieve consensus with a demanding convergent speed through agents' interactions. Numerical examples are given to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the time-domain Maxwell’s equations coupled to another partial differential equation, which arises from modeling of light and structure interaction at the nanoscale. One major contribution of this paper is that the well-posedness is rigorously justified for the first time. Then we propose a fully-discrete finite element method to solve this model. It is interesting to note that we need use curl conforming, divergence conforming, and L 2 finite elements for this model. Numerical stability and optimal error estimate of the scheme are proved. Numerical results justifying our theoretical convergence rate are presented. [ABSTRACT FROM AUTHOR]

Three steps modified Levenberg–Marquardt method for nonlinear equations was introduced by Yang (2013). This method uses the addition of the Levenberg–Marquardt (LM) step and two approximate LM steps as the trial step at every iteration. Using trust region technique, the global and biquadratic convergence of the method is proved by Yang. The main aim of this paper is to introduce a new line search strategy while investigating the convergence properties of the method with this line search technique. Since the search direction of Yang method may be not a descent direction, standard line searches cannot be used directly. In this paper we propose a new nonmonotone third order Armijo type line search technique which guarantees the global convergence of this method while we use an adaptive LM parameter. It is proved that the convergence order of the new method is biquadratic. Numerical results show the new algorithm is efficient and promising. [ABSTRACT FROM AUTHOR]

In this paper, by constructing an objective function and using the gradient search, full-rank and reduced-rank gradient-based algorithms are suggested for solving generalized coupled Sylvester matrix equations. It is proved that the reduced-rank iterative algorithm is convergent for proper initial iterative values. By analyzing the spectral radius of the related matrices, the convergence properties are studied and the optimal convergence factor of the reduced-rank algorithm is determined. The relationship between the reduced-rank algorithm and the full-rank algorithm is discussed. Consequently, the computation load can be reduced greatly for solving a class of matrix equation. A numerical example is provided to illustrate the effectiveness of the proposed algorithms and testify the conclusions suggested in this paper. [ABSTRACT FROM AUTHOR]

The paper studies the convergence of some parallel multisplitting block iterative methods for the solution of linear systems arising in the numerical solution of Euler equations. Some sufficient conditions for convergence are proposed. As special cases the convergence of the parallel block generalized AOR (BGAOR), the parallel block AOR (BAOR), the parallel block generalized SOR (BGSOR), the parallel block SOR (BSOR), the extrapolated parallel BAOR and the extrapolated parallel BSOR methods are presented. Furthermore, the convergence of the parallel block iterative methods for linear systems with special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper. [ABSTRACT FROM AUTHOR]

To avoid finding the stationary distributions of stochastic differential equations by solving the nontrivial Kolmogorov–Fokker–Planck equations, the numerical stationary distributions are used as the approximations instead. This paper is devoted to approximate the stationary distribution of the underlying equation by the Backward Euler–Maruyama method. Currently existing results (Mao et al., 2005; Yuan et al., 2005; Yuan et al., 2004) are extended in this paper to cover larger range of nonlinear SDEs when the linear growth condition on the drift coefficient is violated. [ABSTRACT FROM AUTHOR]

This paper represents the continuation of the analysis from papers Milošević (2011) [10] and Milošević (2013) [11]. The main aim of this paper is to establish certain results for the backward Euler method for a class of neutral stochastic differential equations with time-dependent delay. For that purpose, the split-step backward Euler method, which represents an extension of the backward Euler method, is introduced for this class of equations. Conditions under which the split-step backward Euler method, and thus the backward Euler method, is well defined are revealed. Moreover, the convergence in probability of the backward Euler method is proved under certain nonlinear growth conditions including the one-sided Lipschitz condition. This result is proved using the technique which is based on the application of the continuous-time approximation. For this reason, the discrete forward–backward Euler method is involved since it allows its continuous version to be well defined from the aspect of measurability. The convergence in probability is established for the continuous forward–backward Euler solution, which is essential for proving the same result for both discrete forward–backward and backward Euler methods. Additionally, it is proved that the discrete backward Euler equilibrium solution is globally a.s. asymptotically exponentially stable, without the linear growth condition on the drift coefficient of the equation. As usual, the whole consideration is affected by the presence and properties of the delay function. [ABSTRACT FROM AUTHOR]

In this paper, the steady-state response for a thermoelastic half-plane with voids is studied, in which, the surface of the half-plane is partly subjected to a surface harmonic force and a thermal source. The semi-analytical solutions and the numerical solutions of the half-plane problems with three different materials are obtained from a semi-analytical method and a developed differential quadrature element method in this paper, respectively. The corresponding numerical results are compared and the effects of parameters are considered as well. It can be seen that the semi-analytical solutions and corresponding numerical solutions coincide with each other. This means that the differential quadrature element method is a very efficient method for seeking the numerical solutions of the half-plane problems with discontinuity, and it has some advantageous properties, such as small computational amount, high accuracy, and better convergence. [ABSTRACT FROM AUTHOR]

Abstract: Targeting a new multiple zero finder, in this paper, we suggest an efficient two-point class of methods, when the multiplicity of the root is known. The theoretical aspects are investigated and show that each member of the contributed class achieves fourth-order convergence by using three functional evaluations per full cycle. We also employ numerical examples to evaluate the accuracy of the proposed methods by comparison with other existing methods. For functions with finitely many real roots in an interval, relatively little literature is known, while in applications, the users wish to find all the real zeros at the same time. Hence, the second aim of this paper will be presented by designing a fourth-order algorithm, based on the developed methods, to find all the real solutions of a nonlinear equation in an interval using the programming package Mathematica 8. [Copyright &y& Elsevier]

Abstract In this paper, we present an incremental pressure correction finite element method for the time-dependent Oldroyd flows. This method is a fully discrete projection method. As we all know, most projection methods have been studied without space discretization. Then the ensuing analysis may not extend to this case. We also give the stability analysis and the optimal error analysis. The analysis is based on a time discrete error and a spatial discrete error. In order to show the effectiveness of the method, we also present some numerical results. The numerical results confirm our analysis and show clearly the stability and optimal convergence of the incremental pressure correction finite element method for the time-dependent Oldroyd flows. [ABSTRACT FROM AUTHOR]

Abstract: The goal of this paper is to investigate a backward parabolic equation with time-dependent coefficient by the truncation method (TM). The key point to our paper is that we give an a posteriori choice of regularization parameter for the TM, with which the convergence estimates between the exact solution and the regularized approximation are obtained. Numerical implementation sheds light on the accuracy and efficiency of the proposed method. [Copyright &y& Elsevier]

Abstract: We consider a numerical scheme for the one dimensional time dependent Hamilton–Jacobi equation in the periodic setting. This scheme consists in a semi-discretization using monotone approximations of the Hamiltonian in the spacial variable. From classical viscosity solution theory, these schemes are known to converge. In this paper we present a new approach to the study of the rate of convergence of the approximations based on the nonlinear adjoint method recently introduced by L.C. Evans. We estimate the rate of convergence for convex Hamiltonians and recover the convergence rate in terms of the norm and in terms of the norm, where h is the size of the spacial grid. We discuss also possible generalizations to higher dimensional problems and present several other additional estimates. The special case of quadratic Hamiltonians is considered in detail in the end of the paper. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study a finite difference method for quasilinear coupled problems of partial differential equations that presents numerically an unexpected second order convergence rate. The error analysis presented allows us to conclude that the finite difference method is supraconvergent. As the method studied in this paper can be seen as a fully discrete piecewise linear finite element method, we conclude the supercloseness of our approximations. [Copyright &y& Elsevier]

Abstract: This paper deals with a family of balanced implicit methods with linear interpolation for the stochastic pantograph equations with jumps. In this paper, the strong mean-square convergence theory is established for the numerical solutions of the system. It is shown that the balanced implicit methods, which are fully implicit methods, give strong convergence rate of at least 1/2. For a linear scalar test equation, the balanced implicit methods are shown to capture the mean-square stability for all sufficiently small time-steps under appropriate conditions. Furthermore, weak variants are also considered and their mean-square stability analyzed. Several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean-square stabilities. [Copyright &y& Elsevier]

Abstract: Computational fluid dynamics (CFD) has been introduced to indoor environment study for decades. Analysis of CFD results will first require the reach of a grid-independent CFD solution to eliminate the false information induced by numerical causes, e.g., grid and numerical scheme. Judging a grid-independent solution has been mostly experience-based, leading the necessity of developing an objective and easy-to-use criterion for grid independency evaluation. This paper presents a new approach to assessing the grid independency of CFD modeling for indoor environment applications. A normalized root mean square error (RMSE) index is developed from the grid convergence index (GCI) concept in literature to evaluate the overall differences of predicted results with varying grid resolutions. The paper further introduces the use of numerical viscosity analysis method to verify the grid-independent solution of a CFD analysis, which also provides fundamental insight into the grid-induced error in numerical solutions. Although initially developed for indoor environment applications, the proposed RMSE index along with the numerical viscosity analysis method can be applied to most general CFD studies and has great potential of being incorporated into commercial CFD software to provide user a more accurate and objective alternative of checking grid independency of CFD simulation. [Copyright &y& Elsevier]

Abstract: Two similar stability notions are considered; one is the long established notion of convergent systems, the other is the younger notion of incremental stability. Both notions require that any two solutions of a system converge to each other. Yet these stability concepts are different, in the sense that none implies the other, as is shown in this paper using two examples. It is shown under what additional assumptions one property indeed implies the other. Furthermore, this paper contains necessary and sufficient characterizations of both properties in terms of Lyapunov functions. [Copyright &y& Elsevier]

Abstract: Spectral elements methods provide very accurate solutions of elliptic problems. In this paper we apply the method to the A N (i.e. SP2N−1) approximation of neutron transport. Numerical results for classical benchmark cases highlight its performance in comparison with finite element computations, in terms of accuracy per degree of freedom and convergence rate. All calculations presented in this paper refer to two-dimensional problems. The method can easily be extended to three-dimensional cases. The results illustrate promising features of the method for more complex transport problems. [Copyright &y& Elsevier]

Abstract: In this paper, we are going to consider the following problemwith corresponding measured data functions , a given function f is known as the “source function”. We want to determine the solution for from the Cauchy data at . The problem is ill-posed, as the solution exhibits unstable dependence on the given data functions. The aim of this paper is to introduce new efficient regularization methods, such as truncation of high frequency and quasi-boundary-type methods, with explicit error estimates for an extended case (i.e. the inhomogeneous problem with in Eq. (1)). In addition, we also carry out numerical experiments and compare numerical results of our methods with Qin and Wei’s methods [1] in homogeneous case, as well as to compare our numerical solutions with exact solutions in inhomogeneous case. It shows that our truncation and Qin and Wei’s truncation methods giving almost same results. However, our quasi-boundary-type methods show a better results than quasi-reversibility method of Qin and Wei in term of error estimation and convergence speed. [Copyright &y& Elsevier]

Abstract: We establish in this paper the optimization model of group consensus of 2-tuple linguistic preferential relations (LPRGCO Model), put forward three kinds of solutions to this model, and discover in it the convergence of group consensus. To detect the LPRGCO Model, we first build two kinds of optimal matrices as standards to measure the group consensus of 2-tuple linguistic preference relations (LPRs). And to analyze consensus deviations, we then adopt three types of measures, namely, the individual degree of consistency regarding alternative decision pairs, the deviational degree of the group consensus regarding alternative decision pairs, and the degree of group consensus regarding the original 2-tuple LPRs. On the basis of the previous analysis we not only construct an optimization model to probe into the deviation of the group consensus of 2-tuple LPRs by minimizing the weighted arithmetic average of deviation degrees of individual consistency, but also point out three feasible solutions to this optimization model: the optimal solution, satisfactory solutions, and non-inferior solutions. Accordingly, we discover different conditions in terms of the three solutions. And hence, we can from the aforementioned discussion draw a conclusion that the deviation of group consensus either decreases or stays invariant as the number of decision makers (DM) increases. To expatiate on the practical value of the model proposed, we will display in this paper numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the -stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation. [Copyright &y& Elsevier]

Abstract: This paper improves on generalized properties of a family of iterative methods to compute the approximate inverses of square matrices originally proposed in [1]. And while the methods of [1] can be used to compute the inner inverses of any matrix, it has not been proved that these sequences converge (in norm) to a fixed inner inverse of the matrix. In this paper, it is proved that the sequences indeed are convergent to a fixed inner inverse of the matrix which is the Moore–Penrose inverse of the matrix. The convergence proof of these sequences is given by fundamental matrix calculus, and numerical experiments show that the third-order iterations are as good as the second-order iterations. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and consider a new class of equilibrium variational inequalities, called the mixed general equilibrium bifunction variational inequalities. We suggest and analyze some proximal methods for solving mixed general equilibrium bifunction variational inequalities using the auxiliary principle technique. Convergence of these methods is considered under some mild suitable conditions. Several cases are also discussed. Results in this paper include some new and known results as special cases. [Copyright &y& Elsevier]

Abstract: This paper studies almost-periodic neural networks of Hopfield type described by delayed differential equations. The authors introduce an exponential ordering to analyze the long term behavior of the solutions. They prove some theorems of global convergence and deduce the stabilization role of the fast inhibitory self-connections. The proof, which in the case of the neural networks considered in this paper requires uniform stability, uses arguments of comparison of differential equations and methods of the non-autonomous monotone theory of dynamical systems. When the connections between different neurons are excitatory, improved conditions of convergence are obtained and the stabilization role of strongly positive inputs is also shown. The applicability of the results is illustrated with several numerical experiments based on two different families of neural networks. [Copyright &y& Elsevier]

Abstract In this paper, we consider a coupled system of m (≥ 2) singularly perturbed semilinear reaction–diffusion equations with a discontinuous source term having a discontinuity at a point in the interior of the domain. The diffusion term of each equation is multiplied by small singular perturbation parameter, but these parameters are assumed to be different in magnitude. A numerical method is constructed on a variant of Shishkin mesh. The approximations generated by this method are shown to be almost second order uniformly convergent with respect to all perturbation parameters. Numerical results are in support of the theoretical results. Highlights • A coupled system of m (≥ 2) singularly perturbed semilinear reaction-diffusion equations is considered. • The source term is having a discontinuity at a point in the interior of the domain. • The diffusion term of each equation is multiplied by a small perturbation parameter. • All the perturbation parameters are different in magnitude. • The constructed numerical method has almost second parameter uniform convergence. [ABSTRACT FROM AUTHOR]

Highlights • A type of extended one-leg methods are constructed for a class of nonlinear stiff fractional differential equations. • Under some suitable conditions, the extended one-leg methods are proved to be stable and convergent of order min { p , 2 − γ }. • Several interesting numerical examples are presented to illustrate the computational efficiency and accuracy of the extended one-leg methods. Abstract This paper is concerned with numerical solutions for a class of nonlinear stiff fractional differential equations (SFDEs). By combining the underlying one-leg methods with piecewise linear interpolation, a type of extended one-leg methods for nonlinear SFDEs with γ -order (0 < γ < 1) Caputo derivatives are constructed. It is proved under some suitable conditions that the extended one-leg methods are stable and convergent of order min { p , 2 − γ } , where p is the consistency order of the underlying one-leg methods. Several numerical examples are given to illustrate the computational efficiency and accuracy of the methods. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a preconditioned two-step modulus-based matrix splitting iteration method for linear complementarity problems associated with an M -matrix is proposed. The convergence analysis of the presented method is given. In particular, we provide a comparison theorem between preconditioned two-step modulus-based Gauss–Seidel (PTMGS) iteration method and two-step modulus-based Gauss–Seidel (TMGS) iteration method, which shows that PTMGS method improves the convergence rate of original TMGS method for linear complementarity problem. Numerical tested examples are used to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract Research on the theories and efficient numerical methods of M-matrix algebraic Riccati equation (MARE) has become a hot topic in recent years. In this paper, we consider numerical solution of M-matrix algebraic Riccati equation and propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. Convergence of the MALI method is proved by choosing proper parameters for the nonsingular M-matrix or irreducible singular M-matrix. Theoretical analysis and numerical experiments show that the MALI method is effective and efficient in some cases. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the two MAC schemes are introduced and analyzed to solve the time fractional Stokes equation on non-uniform grids. One is the standard MAC scheme and another is the efficient MAC scheme, where the fast evaluation of the Caputo fractional derivative is used. The stability results are derived. We obtain the second order superconvergence in discrete L 2 norm for both velocity and pressure. We also obtain the second order superconvergence for some terms of the H 1 norm of the velocity on non-uniform grids. Besides, the efficient algorithm for the evaluation of the Caputo fractional derivative is used to save the storage and computation cost greatly. Finally, some numerical experiments are presented to show the efficiency and accuracy of MAC schemes. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a conservative compact finite difference scheme is presented for solving the N-coupled nonlinear Schrödinger–Boussinesq equations. By using the discrete energy method, it is proved that our scheme is unconditionally convergent in the maximum norm and the convergent rate is at O (τ 2 + h 4) with time step τ and mesh size h. Numerical results including the comparisons with other numerical methods are reported to demonstrate the accuracy and efficiency of the method and to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract The spectral conjugate gradient method is an effective method for large-scale unconstrained optimization problems. In this paper, based on Quasi-Newton direction and Quasi-Newton equation, a new spectral conjugate gradient method is proposed. This method is motivated by the three-term modified Polak-Ribiyre-Polyak (PRP) method and spectral parameters. The global convergence of algorithm is proved for general functions under a strong Wolfe line search. Numerical results show that the new algorithm is superior to the three-term modified PRP method. [ABSTRACT FROM AUTHOR]

Abstract In this paper, our main aim is to investigate the stability and strong convergence of an implicit numerical approximations for neutral-type stochastic differential equations with superlinearly growing coefficients. After providing moment boundedness and exponential stability for the exact solutions, we show that the backward Euler–Maruyama numerical method preserves stability and boundedness of moments, and the numerical approximations converge strongly to the true solutions for sufficiently small step size. [ABSTRACT FROM AUTHOR]

Highlights • A new variational model for image retinex is proposed. • The existence of minimizer for the variational model is proved. • An efficient algorithm is derived to solve the model by adding auxiliary variables. • The convergence of the numerical algorithm is proved under some assumptions. • The proposed method can be extended to other regularization and fidelity terms. Abstract In this paper, we propose a detail preserving variational model for Retinex to simultaneously estimate the illumination and the reflectance from an observed image. Most previous models use the log-transform as pretreatment which results in loss of details in reflectance. From this observation, a detail preserving variational method is proposed for better decomposition. Different from the log-transform based models, the proposed model performs the decomposition directly in the image domain. Mathematically, we prove the existence of a solution for the proposed model. Numerically, we derive an efficient iterative algorithm by utilizing alternating direction method of multipliers (ADMM) method. Experimental results demonstrate the effectiveness of the proposed method. Compared with other closely related Retinex methods, the proposed method achieves competitive results on both subjective and objective assessments. [ABSTRACT FROM AUTHOR]

Abstract We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by further study on the generating function of the discretization matrix, second order convergence of the scheme is proved for diffusion coefficients satisfying a certain condition but are not necessary to be proportional. The theoretical results are justified by numerical tests. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we discuss the time-dependent generalized membrane shell model, which has not been addressed numerically in literature. We show that the solution of this model exists and is unique. We first provide a numerical method for the time-dependent generalized membrane shell. Concretely, we semi-discretize the space variable and fully discretize the problem using time discretization by the Newmark scheme. The corresponding numerical analyses of existence, uniqueness, stability and convergence with a priori error estimates are given. Finally, we present numerical experiments with a portion of the conical shell and a portion of the hyperbolic shell to verify theoretical convergence results and demonstrate the effectiveness of the numerical scheme. Highlights • Existence and uniqueness of the time-dependent generalized membrane shell. • Newmark scheme for the time-dependent generalized membrane shell. • Analyses of existence, uniqueness, stability, convergence and a priori error estimate for elastodynamic problem. [ABSTRACT FROM AUTHOR]

Highlights • A fast successive relaxation updating method for continuous-discontinuous cellular automaton(CDCA) is proposed. • A fast CDCA is developed, and increments of displacement and nodal force are enlarged by the accelerating factor. • A new discontinuity tracking method which combines cell space cutting and cell neighbor searching is proposed. • The optimal value of the accelerating factor is studied, and an adaptive iteration scheme is proposed. Abstract Because of its local property, cellular automaton method has been widely applied in different subjects, but the main problem is that the cellular updating is time-consuming. In order to improve its calculation efficiency, a fast successive relaxation updating method is proposed in this paper. Firstly, an accelerating factor ω is defined, and a fast successive relaxation updating theory and its corresponding convergence conditions are developed. In each updating step, the displacement increment is enlarged ω times as a new increment to replace the old one, similarly, the nodal forces for its neighbors caused by this displacement increment are also enlarged by the same accelerating factor, and do those updating operations until the convergence is achieved. By this method, the convergence rate is greatly improved, by a suitable accelerating factor, 90 to 98% of iteration steps are decreased compared to that of the traditional method. Besides, the influence factors for the accelerating factor are studied, and numerical studies show that the suitable accelerating factor is 1.85 < ω < 1.99, which is greatly influenced by cell stiffness, and the optimal accelerating factor is 1.96 < ω < 1.99. Finally, numerical examples are given to illustrate that the present method is effective and high convergence rate compared to the traditional method. [ABSTRACT FROM AUTHOR]

Highlights • A penalty finite element method for the steady MHD equations was given. • The solution of the penalty method convergence the solution of the steady MHD equations. • The stability analysis shows our method is stable. • The error estimate shows our method has an optimal convergence order. • The numerical results of the Hartmann flow was shown. Abstract In this paper, we give a penalty finite element method for the steady MHD equations. In this method, we decouple the MHD into two equations, one for the velocity and magnetic (u , B) , the other for the pressure p. We prove the existence of the penalty method and the optimal error estimate. Then, we give the penalty finite element method for the MHD equations. The stability analysis shows our method is stable, and the error estimate shows our method has an optimal convergence order. Finally, we give some numerical results of exact solution equation and Hartmann flow. The numerical results show that our penalty finite element method is effect. [ABSTRACT FROM AUTHOR]

This paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one-sided Lipschitz drift. Strong convergence rate is revealed under a global one-sided Lipschitz condition, while for a local one-sided Lipschitz condition, the tamed theta scheme is modified to ensure the well-posedness of implicit numerical schemes, then we show the convergence of the numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper, by adapting the underlying implicit-explicit (IMEX) one-leg methods (cf. [1, 2]), a class of extended IMEX one-leg (EIEOL) methods are suggested for solving nonlinear stiff neutral equations (SNEs). It is proven under some suitable conditions that EIEOL methods are D-convergent of order 2 and stable for nonlinear SNEs. Several numerical examples are given to testify the obtained theoretical results and the computational effectiveness of EIEOL methods. Moreover, a comparison with the fully implicit one-leg methods is presented, which shows that EIEOL methods have the higher computational efficiency. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the superconvergence analysis of a two-grid method (TGM) is established for the semilinear parabolic equations. Based on the combination of the interpolation and Ritz projection technique, an important ingredient in the method, the superclose estimates in the H 1 -norm are deduced for the backward Euler fully-discrete TGM scheme. Moreover, through the interpolated postprocessing approach, the corresponding global superconvergence result is derived. Finally, some numerical results are provided to confirm the theoretical analysis, and also show that the computing cost of the proposed TGM is only half of the conventional Galerkin finite element methods (FEMs). [ABSTRACT FROM AUTHOR]

In this paper, a multiobjective hybrid bat algorithm is proposed to solve the combined economic/emission dispatch problem with power flow constraints. In the proposed algorithm, an elitist nondominated sorting method and a modified crowding-distance sorting method are introduced to acquire an evenly distributed Pareto Optimal Front. A modified comprehensive learning strategy is used to enhance the learning ability of population. Through this way, each individual can learn not only from all individual best solutions but also from the global best solutions (nondominated solutions). A random black hole model is introduced to ensure that each dimension in current solution can be updated individually with a predefined probability. This is not only meaningful in enhancing the global search ability and accelerating convergence speed, but particularly key to deal with high dimensional systems, especially large-scale power systems. In addition, chaotic map is integrated to increase the diversity of population and avoid premature convergence. Finally, numerical examples on the IEEE 30-bus, 118-bus and 300-bus systems, are provided to demonstrate the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Full waveform inversion (FWI) is an efficient engine for estimating subsurface model parameters. FWI is often implemented via local optimization methods such as conjugate gradients or steepest descent. In this paper, we examine the effect of model parameterization on the estimation of velocity models obtained via Newton-based optimization FWI methods. We consider three model parameterizations for acoustic FWI. These include velocity, slowness and slowness squared parameterizations. We numerically demonstrate that model parametrization significantly influences the convergence rate of the Newton method. A high convergence rate for Newton-based FWI algorithms can be attained by using a slowness squared model parameterization. The latter is true for data with relatively high signal-to-noise ratio. However, in the case of data contaminated by high levels of noise, the slowness parameterization provides a good trade-off between the resolution of the reconstructed model parameters and convergence rate in comparison to velocity or slowness squared parameterization. Numerical results were conducted with the Marmousi velocity model to highlight our premises. [ABSTRACT FROM AUTHOR]

Abstract Regenerative chatter has a negative effect on the quality of machined surfaces in milling and it is thus vital to predict this accurately. The full-discretization method (FDM) is extensively utilized to predict the chatter stability. However, the effects of helix angle and multi-mode usually cause poor prediction by FDM and are often ignored. The existing fourth-order FDM has better computational efficiency and convergence rate than other existing FDMs. In this paper, a fourth-order FDM is optimized to improve prediction accuracy by considering both the helix angle and multi-mode. Based on the numerical stability computation results, it can firstly be observed that the stability lobe diagram (SLD) obtained using the proposed FDM has advantages over the existing SDM in the aspects of both the convergence rate and computational efficiency, and the effect of helix angle on the SLD is determined by both the number of teeth, radial immersion ratio and helix angle of the tool. Furthermore, a detailed investigation on how to detect chatter in milling experiments is made using an integrated method with time-frequency analysis. Finally, comparing the prediction result with the experimental milling results, it can be concluded that the proposed SLD with first three modes excels in prediction accuracy. A quantitative evaluation about the improvement of prediction accuracy is also made. The intention of proposing the updated FDM is to make the chatter prediction more accurate and efficient. Highlights • The proposed FDM considers the helix angle and multi-mode effects. • FDM with 4th-order interpolation for DDEs' state term is utilized for optimization. • An integrated method with time-frequency analysis is proposed to detect chatter. • Improved precision in chatter stability prediction in milling is demonstrated. [ABSTRACT FROM AUTHOR]