Authors:Yunying Zheng; Zhengang Zhao Pages: 32 - 41 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): Yunying Zheng, Zhengang Zhao The cable equation as one of the best models for simulating neurodynamics can be derived from the Nernst–Planck equation which simulates the electrodiffusion of ions. Recently, some researchers find that in nerve cells molecular diffusion is anomalous subdiffusion. It is much more effective using fractional cable equation for simulating the dynamic behavior. In this paper, by introducing an auxiliary function w = ∂ u / ∂ t , the fractional cable equation can be changed into a system of integro-differential equations. Then a full discrete numerical method for solving the system is studied, where in time axis the discontinuous Galerkin finite element method is used and in spacial axis the Galerkin finite element scheme is adopted. The existence and uniqueness of the numerical solution are included. The convergence is also discussed in detail. Numerical examples are also included to demonstrate the effectiveness of the theoretical results.

Authors:Oleg V. Motygin Abstract: Publication date: Available online 7 January 2017 Source:Applied Numerical Mathematics Author(s): Oleg V. Motygin Green's function of the problem describing steady forward motion of bodies in an open ocean in the framework of the linear surface wave theory (the function is often referred to as Kelvin's wave source potential) is considered. Methods for numerical evaluation of the so-called ‘single integral’ (or, in other words, ‘wavelike’) term, dominating in the representation of Green's function in the far field, are developed. The difficulty in the numerical evaluation is due to integration over infinite interval of the function containing two differently oscillating factors and the presence of stationary points. This work suggests two methods to approximate the integral. First of them is based on the idea put forward by D. Levin in 1982 — evaluation of the integral is converted to finding a particular slowly oscillating solution of an ordinary differential equation. To overcome well-known numerical instability of Levin's collocation method, an alternative type of collocation is used; it is based on a barycentric Lagrange interpolation with a clustered set of nodes. The second method for evaluation of the wavelike term involves application of the steepest descent method and Clenshaw–Curtis quadrature. The methods are numerically tested and compared.

Authors:Guanyu Zhou Abstract: Publication date: Available online 9 January 2017 Source:Applied Numerical Mathematics Author(s): Guanyu Zhou We consider the fictitious domain method with penalty for the parabolic problem in a moving-boundary domain. Two types of penalty (the H 1 and L 2 -penalty methods) are investigated, for which we obtain the error estimate of penalty. Moreover, for H 1 -penalty method, the H 2 -regularity and a-priori estimate depending on the penalty parameter ϵ are obtained. We apply the finite element method to the H 1 -penalty problem, and obtain the stability and error estimate for the numerical solution. The theoretical results are confirmed by the numerical experiments.

Authors:K. Maleknejad; A. Ostadi Abstract: Publication date: Available online 9 January 2017 Source:Applied Numerical Mathematics Author(s): K. Maleknejad, A. Ostadi In this paper, efficient and computationally attractive methods based on the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations for the numerical solution of a system of Volterra integral equations with weakly singular kernels are presented. Simplicity for performing even in the presence of singularities is one of the advantages of Sinc methods. Convergence analysis of the proposed methods is given and an exponential convergence is achieved as well. Numerical results are presented which demonstrate the efficiency and high accuracy of the proposed methods.

Authors:Wen Li; Guohui Song; Guangming Yao Abstract: Publication date: Available online 10 January 2017 Source:Applied Numerical Mathematics Author(s): Wen Li, Guohui Song, Guangming Yao The standard moving least squares (MLS) method might have an expensive computational cost when the number of test points and the dimension of the approximation space are large. To reduce the computational cost, this paper proposes a piece-wise moving least squares approximation method (PMLS) for scattered data approximation. We further apply the PMLS method to solve time-dependent partial differential equations (PDE) numerically. It is proven that the PMLS method is an optimal design with certain localized information. Numerical experiments are presented to demonstrate the efficiency and accuracy of the PMLS method in comparison with the standard MLS method in terms of accuracy and efficiency.

Authors:Bhupen Deka Abstract: Publication date: Available online 12 January 2017 Source:Applied Numerical Mathematics Author(s): Bhupen Deka In this article a fitted finite element method is proposed and analyzed for wave equation with discontinuous coefficients. Typical semidiscrete and an implicit fully discrete schemes are presented and analyzed. Optimal a priori error estimates for both semi-discrete and fully discrete scheme are proved in L ∞ ( L 2 ) norm. The convergence analysis relies heavily on time reconstructions of continuous and discrete solutions, in conjunction with some known results on elliptic interface problems. Finally, a numerical experiment is presented to verify our theoretical result.

Authors:Charles Puelz; Sunčica Čanić; Béatrice Rivière; Craig G. Rusin Abstract: Publication date: Available online 11 January 2017 Source:Applied Numerical Mathematics Author(s): Charles Puelz, Sunčica Čanić, Béatrice Rivière, Craig G. Rusin One–dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we comment on some theoretical differences among models and systematically compare them for physiologically relevant vessel parameters, network topology, and boundary data. In particular, the effect of the velocity profile is investigated in the cases of both smooth and discontinuous solutions, and a recommendation for a physiological model is provided. The models are discretized by a class of Runge–Kutta discontinuous Galerkin methods.

Authors:Wenjie Shi; Chengjian Zhang Abstract: Publication date: Available online 3 January 2017 Source:Applied Numerical Mathematics Author(s): Wenjie Shi, Chengjian Zhang In this paper, the generalized polynomial chaos (gPC) method is extended to solve nonlinear random delay differential equations (NRDDEs). The error estimation of the method is derived, which arises mainly from a finite-dimensional noise assumption, projection error and discretization error. When the error from the finite-dimensional noise assumption can not be omitted, the error of the method converges to a limit inferior which is just the error from the finite-dimensional noise assumption. With some numerical experiments, the obtained theoretical results are further illustrated.

Authors:Kareem T. Elgindy Pages: 1 - 25 Abstract: Publication date: March 2017 Source:Applied Numerical Mathematics, Volume 113 Author(s): Kareem T. Elgindy The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer–Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles [14]. The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.

Authors:Kareem T. Elgindy Pages: 1 - 25 Abstract: Publication date: March 2017 Source:Applied Numerical Mathematics, Volume 113 Author(s): Kareem T. Elgindy The work reported in this article presents a high-order, stable, and efficient Gegenbauer pseudospectral method to solve numerically a wide variety of mathematical models. The proposed numerical scheme exploits the stability and the well-conditioning of the numerical integration operators to produce well-conditioned systems of algebraic equations, which can be solved easily using standard algebraic system solvers. The core of the work lies in the derivation of novel and stable Gegenbauer quadratures based on the stable barycentric representation of Lagrange interpolating polynomials and the explicit barycentric weights for the Gegenbauer–Gauss (GG) points. A rigorous error and convergence analysis of the proposed quadratures is presented along with a detailed set of pseudocodes for the established computational algorithms. The proposed numerical scheme leads to a reduction in the computational cost and time complexity required for computing the numerical quadrature while sharing the same exponential order of accuracy achieved by Elgindy and Smith-Miles [14]. The bulk of the work includes three numerical test examples to assess the efficiency and accuracy of the numerical scheme. The present method provides a strong addition to the arsenal of numerical pseudospectral methods, and can be extended to solve a wide range of problems arising in numerous applications.

Authors:Giuseppe Izzo; Zdzislaw Jackiewicz Pages: 71 - 92 Abstract: Publication date: March 2017 Source:Applied Numerical Mathematics, Volume 113 Author(s): Giuseppe Izzo, Zdzislaw Jackiewicz We investigate implicit–explicit (IMEX) Runge–Kutta (RK) methods for differential systems with non-stiff and stiff processes. The construction of such methods with large regions of absolute stability of the ‘explicit part’ of the method assuming that the ‘implicit part’ of the scheme is A-stable, is described. We also describe the construction of IMEX RK methods, where the ‘explicit part’ of the schemes have strong stability properties. Examples of highly stable IMEX RK methods are provided up to the order p = 4 . Numerical examples are also given which illustrate good performance of these schemes.

Authors:Alberto Crivellaro; Simona Perotto; Stefano Zonca Pages: 93 - 108 Abstract: Publication date: March 2017 Source:Applied Numerical Mathematics, Volume 113 Author(s): Alberto Crivellaro, Simona Perotto, Stefano Zonca We propose new algorithms to overcome two of the most constraining limitations of surface reconstruction methods in use. In particular, we focus on the large amount of data characterizing standard acquisitions by scanner and the noise intrinsically introduced by measurements. The first algorithm represents an adaptive multi-level interpolating approach, based on an implicit surface representation via radial basis functions. The second algorithm is based on a least-squares approximation to filter noisy data. The third approach combines the two algorithms to merge the correspondent improvements. An extensive numerical validation is performed to check the performances of the proposed techniques.

Authors:Jinbiao Wu; Hui Zheng Pages: 109 - 123 Abstract: Publication date: March 2017 Source:Applied Numerical Mathematics, Volume 113 Author(s): Jinbiao Wu, Hui Zheng In this paper we study the multigrid methods for adaptively refined finite element meshes. In our multigrid iterations, on each level we only perform relaxation on new nodes and the old nodes whose support of nodal basis function have changed. The convergence analysis of the algorithm is based on the framework of subspace decomposition and subspace correction. In order to decompose the functions from the finest finite element space into each level, a new projection is presented in this paper. Briefly speaking, this new projection can be seemed as the weighted average of the local L 2 projection. We can perform our subspace decomposition through this new projection by its localization property. Other properties of this new projection are also presented and by these properties we prove the uniform convergence of the algorithm in both 2D and 3D. We also present some numerical examples to illustrate our conclusion.

Authors:Mahboub Baccouch Pages: 124 - 155 Abstract: Publication date: March 2017 Source:Applied Numerical Mathematics, Volume 113 Author(s): Mahboub Baccouch In this paper, we investigate the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the two-dimensional sine-Gordon nonlinear hyperbolic equation on Cartesian grids. We perform a local error analysis and show that the actual error can be split into an O ( h p + 1 ) leading component and a higher-order component, when tensor product polynomials of degree at most p are used. We further prove that the leading term of the LDG error is spanned by two ( p + 1 ) -degree Radau polynomials in the x and y directions, respectively. Thus, the LDG solution is O ( h p + 2 ) superconvergent at Radau points obtained as a tensor product of the roots of ( p + 1 ) -degree right Radau polynomial. Computational results indicate that our superconvergence results hold globally. We use these results to construct simple, efficient, and asymptotically exact a posteriori LDG error estimates. The proposed error estimates are computationally simple and are obtained by solving local steady problems with no boundary conditions on each element. Finally, we present several numerical examples to validate the superconvergence results and the asymptotic exactness of our a posteriori error estimates under mesh refinement.

Authors:J.B. Francisco; F.S. Viloche Bazán; M. Weber Mendonça Pages: 51 - 64 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): J.B. Francisco, F.S. Viloche Bazán, M. Weber Mendonça This paper concerns a non-monotone algorithm for minimizing differentiable functions on closed sets. A general numerical scheme is proposed which combines a regularization/trust-region framework with a non-monotone strategy. Global convergence to stationary points is proved under usual assumptions. Numerical experiments for a particular version of the general algorithm are reported. In addition, a promising numerical scheme for medium/large-scale orthogonal Procrustes problem is also proposed and numerically illustrated.

Authors:Qiangqiang Zhu; Zhen Gao; Wai Sun Don; Xianqing Lv Pages: 65 - 78 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): Qiangqiang Zhu, Zhen Gao, Wai Sun Don, Xianqing Lv We investigate the performance of the high order well-balanced hybrid compact-weighted essentially non-oscillatory (WENO) finite difference scheme (Hybrid) for simulations of shallow water equations with source terms due to a non-flat bottom topography. The Hybrid scheme employs the nonlinear fifth order characteristic-wise WENO-Z finite difference scheme to capture high gradients and discontinuities in an essentially non-oscillatory manner, and the linear spectral-like sixth order compact finite difference scheme to resolve the fine scale structures in the smooth regions of the solution efficiently and accurately. The high order multi-resolution analysis is employed to identify the smoothness of the solution at each grid point. In this study, classical one- and two-dimensional simulations, including a long time two-dimensional dam-breaking problem with a non-flat bottom topography, are conducted to demonstrate the performance of the hybrid scheme in terms of the exact conservation property (C-property), good resolution and essentially non-oscillatory shock capturing of the smooth and discontinuous solutions respectively, and up to 2–3 times speedup factor over the well-balanced WENO-Z scheme.

Authors:A. Shoja; A.R. Vahidi; E. Babolian Pages: 79 - 90 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): A. Shoja, A.R. Vahidi, E. Babolian In this paper, a spectral iterative method is employed to obtain approximate solutions of singular nonlinear Volterra integral equations, called Abel type of Volterra integral equations. The Abel's type nonlinear Volterra integral equations are reduced to nonlinear fractional differential equations. This approach is based on a combination of two different methods, i.e. the iterative method proposed in [7] and the spectral method. The method reduces the fractional differential equations to systems of linear algebraic equations and then the resulting systems are solved by a numerical method. Finally, we prove that the spectral iterative method (SIM) is convergent. Numerical results comparing this iterative approach with alternative approaches offered in [4,8,24] are presented. Error estimation also corroborate numerically.

Authors:Vu Thai Luan Pages: 91 - 103 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): Vu Thai Luan Among the family of fourth-order time integration schemes, the two-stage Gauss–Legendre method, which is an implicit Runge–Kutta method based on collocation, is the only superconvergent. The computational cost of this implicit scheme for large systems, however, is very high since it requires solving a nonlinear system at every step. Surprisingly, in this work we show that one can construct and prove convergence results for exponential methods of order four which use two stages only. Specifically, we derive two new fourth-order two-stage exponential Rosenbrock schemes for solving large systems of differential equations. Moreover, since the newly schemes are not only superconvergent but also fully explicit, they turn out to be very competitive compared to the two-stage Gauss–Legendre method as well as other fourth-order time integration schemes. Numerical experiments are given to demonstrate the efficiency of the new integrators.

Authors:V. Baron; Y. Coudière; P. Sochala Pages: 104 - 125 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): V. Baron, Y. Coudière, P. Sochala We derive some a posteriori error estimates for the Richards equation. This parabolic equation is nonlinear in space and in time, thus its resolution requires fixed-point iterations within each time step. We measure the approximation error with the dual norm of the residual. A computable upper bound of this error consists of several estimators involving adequate reconstructions based on the degrees of freedom of the scheme. The space and time reconstructions are specified for a two-step backward differentiation formula and a discrete duality finite volume scheme. Our strategy to decrease the computational cost relies on an aggregation of the estimators in three components: space discretization, time discretization, and linearization. We propose an algorithm to stop the fixed-point iterations after the linearization error becomes negligible, and to choose the time step in order to balance the time and space errors. We analyze the influence of the parameters of this algorithm on three test cases and quantify the gain obtained in comparison with a classical simulation.

Authors:Yanyan Yu; Weihua Deng; Yujiang Wu; Jing Wu Pages: 126 - 145 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): Yanyan Yu, Weihua Deng, Yujiang Wu, Jing Wu Power-law probability density function (PDF) plays a key role in both subdiffusion and Lévy flights. However, sometimes because of the finiteness of the lifespan of the particles or the boundedness of the physical space, tempered power-law PDF seems to be a more physical choice and then the tempered fractional operators appear; in fact, the tempered fractional operators can also characterize the transitions among subdiffusion, normal diffusion, and Lévy flights. This paper focuses on the finite difference schemes for space tempered fractional diffusion equations, being much different from the ones for pure fractional derivatives. By using the generation function of the matrix and Weyl's theorem, the stability and convergence of the derived schemes are strictly proved. Some numerical simulations are performed to testify the effectiveness and numerical accuracy of the obtained schemes.

Authors:Nélio Henderson; Marroni de Sá Rêgo; Janaína Imbiriba Pages: 155 - 166 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): Nélio Henderson, Marroni de Sá Rêgo, Janaína Imbiriba We apply a recently revisited version of the topographical global initialization to solve nonlinear systems of equations with multiple roots subject to inequality constraints. This initialization technique is a simple and ingenious approach based on elementary concepts of graph theory. Here, the topographical initialization is used to generate good starting points to solve constrained global minimization problems, whose solutions are roots of associated nonlinear systems. To accomplish the task of local search, in the minimization step we use a well-established interior-point method. Our methodology was compared against other methods using benchmarks from the literature. Results indicated that the present approach is a powerful strategy for finding all roots of nonlinear systems.

Authors:Rong An; Yuan Li Pages: 167 - 181 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): Rong An, Yuan Li This paper focuses on a linearized fully discrete projection scheme for time-dependent magnetohydrodynamics equations in three-dimensional bounded domain. It is shown that the proposed projection scheme allows for a discrete energy inequality and is unconditionally stable. In addition, we present a rigorous analysis for the rates of convergence.

Authors:Kamana Porwal Pages: 182 - 202 Abstract: Publication date: February 2017 Source:Applied Numerical Mathematics, Volume 112 Author(s): Kamana Porwal In this article, we propose and analyze discontinuous Galerkin (DG) methods for a contact problem with Tresca friction for the linearized elastic material. We derive a residual based a posteriori error estimator for the proposed class of DG methods. The reliability and the efficiency of a posteriori error estimator is shown. We further investigate a priori error estimates under the minimal regularity assumption on the exact solution. An important property shared by a class of DG methods, allow us to carry out the analysis in a unified framework. Numerical experiments are reported to illustrate theoretical results.

Authors:Quan Zheng; Xin Zhao; Yufeng Liu Pages: 1 - 16 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Quan Zheng, Xin Zhao, Yufeng Liu This paper studies a finite difference method for one-dimensional nonhomogeneous Burgers' equation on the infinite domain. Two exact nonlinear artificial boundary conditions are applied on two artificial boundaries to limit the original problem onto a bounded computational domain. A function transformation makes both Burgers' equation and artificial boundary conditions linear. Consequently, a novel finite difference scheme is developed by using the method of reduction of order for the obtained equation and artificial boundary conditions. The stability and the convergence with order 3/2 in time and 2 in space in an energy norm are proved for this method for Burgers' equation. Different examples illustrate the unconditional stability and the accuracy of the proposed method.

Authors:Yubo Yang; Peng Zhu Pages: 36 - 48 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Yubo Yang, Peng Zhu In this paper, we introduce discontinuous Galerkin methods with interior penalties, both the NIPG and SIPG method for solving 2D singularly perturbed convection–diffusion problems. On the modified graded meshes with the standard Lagrange Q k -elements ( k = 1 , 2 ), we show optimal order error estimates in the ε-weighted energy norm uniformly, up to a logarithmic factor, in the singular perturbation parameter ε. We prove that the convergence rate in the ε-weighted energy norm is O ( log k + 1 ( 1 ε ) N k ) , where the total number of the mesh points is O ( N 2 ) . For k ≥ 3 , our methods can be extended directly, provided the higher order regularities of the solution u are derived. Finally, numerical experiments support our theoretical results.

Authors:Jason Albright; Yekaterina Epshteyn; Michael Medvinsky; Qing Xia Pages: 64 - 91 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Jason Albright, Yekaterina Epshteyn, Michael Medvinsky, Qing Xia Numerical approximations and computational modeling of problems from Biology and Materials Science often deal with partial differential equations with varying coefficients and domains with irregular geometry. The challenge here is to design an efficient and accurate numerical method that can resolve properties of solutions in different domains/subdomains, while handling the arbitrary geometries of the domains. In this work, we consider 2D elliptic models with material interfaces and develop efficient high-order accurate methods based on Difference Potentials for such problems.

Authors:Ji-Feng Bao; Chong Li; Wei-Ping Shen; Jen-Chih Yao; Sy-Ming Guu Pages: 92 - 110 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Ji-Feng Bao, Chong Li, Wei-Ping Shen, Jen-Chih Yao, Sy-Ming Guu We propose several approximate Gauss–Newton methods, i.e., the truncated, perturbed, and truncated-perturbed GN methods, for solving underdetermined nonlinear least squares problems. Under the assumption that the Fréchet derivatives are Lipschitz continuous and of full row rank, Kantorovich-type convergence criteria of the truncated GN method are established and local convergence theorems are presented with the radii of convergence balls obtained. As consequences of the convergence results for the truncated GN method, convergence theorems of the perturbed and truncated-perturbed GN methods are also presented. Finally, numerical experiments are presented where the comparisons with the standard inexact Gauss–Newton method and the inexact trust-region method for bound-constrained least squares problems [23] are made.

Authors:Zheng Ma; Yong Zhang; Zhennan Zhou Pages: 144 - 159 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Zheng Ma, Yong Zhang, Zhennan Zhou In this paper, we propose a new time splitting Fourier spectral method for the semi-classical Schrödinger equation with vector potentials. Compared with the results in [21], our method achieves spectral accuracy in space by interpolating the Fourier series via the NonUniform Fast Fourier Transform (NUFFT) algorithm in the convection step. The NUFFT algorithm helps maintain high spatial accuracy of Fourier method, and at the same time improve the efficiency from O ( N 2 ) (of direct computation) to O ( N log N ) operations, where N is the total number of grid points. The kinetic step and potential step are solved by analytical solution with pseudo-spectral approximation, and, therefore, we obtain spectral accuracy in space for the whole method. We prove that the method is unconditionally stable, and we show improved error estimates for both the wave function and physical observables, which agree with the results in [3] for vanishing potential cases and are superior to those in [21]. Extensive one and two dimensional numerical studies are presented to verify the properties of the proposed method, and simulations of 3D problems are demonstrated to show its potential for future practical applications.

Authors:Liangliang Sun; Ting Wei Pages: 160 - 180 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Liangliang Sun, Ting Wei This paper is devoted to identify the zeroth-order coefficient in a time-fractional diffusion equation from two boundary measurement data in one-dimensional case. The existence and uniqueness of two kinds of weak solutions for the direct problem with Neumann boundary condition are proved. We provide the uniqueness for recovering the zeroth-order coefficient and fractional order simultaneously by the Laplace transformation and Gel'fand–Levitan theory. The identification of the zeroth-order coefficient is formulated into a variational problem by the Tikhonov regularization. The existence, stability and convergence of the solution for the variational problem are provided. We deduce an adjoint problem and then use a conjugate gradient method to solve the variational problem. Two numerical examples are provided to show the effectiveness of the proposed method.

Authors:Maryam Kamranian; Mehdi Dehghan; Mehdi Tatari Pages: 181 - 196 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Maryam Kamranian, Mehdi Dehghan, Mehdi Tatari A new adaptive moving least squares (MLS) method with variable radius of influence is presented to improve the accuracy of Meshless Local Petrov–Galerkin (MLPG) methods and to minimize the computational cost for the numerical solution of singularly perturbed boundary value problems. An error indicator based on a posteriori error estimation, accurately captures the regions of the domain with insufficient resolution and adaptively determines the new nodes location. The effectiveness of the new method is demonstrated on some singularly perturbed problems involving boundary layers.

Authors:A.H. Bhrawy; M.A. Zaky Pages: 197 - 218 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): A.H. Bhrawy, M.A. Zaky Current discretizations of variable-order fractional (V-OF) differential equations lead to numerical solutions of low order of accuracy. This paper explores a high order numerical scheme for multi-dimensional V-OF Schrödinger equations. We derive new operational matrices for the V-OF derivatives of Caputo and Riemann–Liouville type of the shifted Jacobi polynomials (SJPs). These allow us to establish an efficient approximate formula for the Riesz fractional derivative. An operational approach of the Jacobi collocation approach for the approximate solution of the V-OF nonlinear Schrödinger equations. The main characteristic behind this approach is to investigate a space–time spectral approximation for spatial and temporal discretizations. The proposed spectral scheme, both in temporal and spatial discretizations, is successfully developed to handle the two-dimensional V-OF Schrödinger equation. Numerical results indicating the spectral accuracy and effectiveness of this algorithm are presented.

Authors:F. Guillén-González; M.V. Redondo-Neble Pages: 219 - 245 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): F. Guillén-González, M.V. Redondo-Neble 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 .

Authors:Heping Ma; Yonghui Qin; Qiuli Ou Pages: 246 - 259 Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): Heping Ma, Yonghui Qin, Qiuli Ou The multidomain Legendre–Galerkin Chebyshev-collocation method is considered to solve one-dimensional linear evolution equations with two nonhomogeneous jump conditions. The scheme treats the first jump condition essentially and the second one naturally. We adopt appropriate base functions to deal with interfaces. The proposed method can be implemented in parallel. Error analysis shows that the approach has an optimal convergence rate. The proposed method is also applied to computing the one-dimensional Maxwell equation and the one-dimensional two phase Stefan problem, respectively. Numerical examples are given to confirm the theoretical analysis.

Authors:Francesco A. Costabile; Anna Napoli Abstract: Publication date: Available online 16 December 2016 Source:Applied Numerical Mathematics Author(s): Francesco A. Costabile, Anna Napoli A general procedure to determine collocation methods for high order boundary value problems is presented. These methods provide globally continuous differentiable solution in the form of polynomial functions and also numerical solution on a set of discrete points. Some special cases are considered. Numerical experiments are presented which support theoretical results and provide favorable comparisons with other existing methods.

Authors:Kavita Goyal; Mani Mehra Abstract: Publication date: Available online 9 December 2016 Source:Applied Numerical Mathematics Author(s): Kavita Goyal, Mani Mehra This paper proposes an adaptive meshfree spectral graph wavelet method to solve partial differential equations. The method uses radial basis functions for interpolation of functions and for approximation of the differential operators. It uses multiresolution analysis based on spectral graph wavelet for adaptivity. The set of scattered node points is subject to dynamic changes at run time which leads to adaptivity. The beauty of the method lies in the fact that the same operator is used for the approximation of differential operators and for the construction of spectral graph wavelet. Initially, we have applied the method on spherical diffusion equation. After that the problem of pattern formation on the surface of the sphere (using Turing equations) is addressed to test the strength of the method. The numerical results show that the method can accurately capture the emergence of the localized patterns at all the scales and the node arrangement is accordingly adapted. The convergence of the method is verified. For each test problem, the CPU time taken by the proposed method is compared with the CPU time taken by a traditional method (spectral method using radial basis functions). It is observed that the adaptive meshfree spectral graph wavelet method is highly efficient.

Authors:Zewen Wang; Shufang Qiu Abstract: Publication date: Available online 8 December 2016 Source:Applied Numerical Mathematics Author(s): Zewen Wang, Shufang Qiu In this paper, a numerical method is proposed to approximate the solution of a two-dimensional scattering problem of time-harmonic elastic wave from a rigid obstacle. By Helmholtz decomposition, the scattering problem is reduced to a system of Helmholtz equations with coupled boundary conditions. Then, we prove that the system of Helmholtz equations has only one solution under certain conditions, and propose an integral equation method to solve it numerically based on Tikhonov regularization method. Finally, numerical examples are presented to show the feasibility and effectiveness of the proposed method.

Authors:S. Amiri; S.M. Hosseini Abstract: Publication date: Available online 8 December 2016 Source:Applied Numerical Mathematics Author(s): S. Amiri, S.M. Hosseini In this paper we introduce a family of stochastic Runge–Kutta Rosenbrock (SRKR) type methods for multi-dimensional Itô stochastic differential equations (SDEs). The presented class of semi-implicit methods need less computational effort in comparison with some implicit ones. General order conditions for the coefficients and the random variables of the SRKR methods are obtained. Then a set of order conditions for a subclass of stochastic weak second order is given. Numerical examples are presented to demonstrate the efficiency and accuracy of the new schemes.

Authors:Hassan Majidian Abstract: Publication date: Available online 16 November 2016 Source:Applied Numerical Mathematics Author(s): Hassan Majidian It is well known that the coefficients of the Chebyshev expansion of a function f ∈ C [ − 1 , 1 ] decay at a rate depending on the smoothness of f. New decay rates for the Chebyshev coefficients as well as their partial sums are obtained which are sharper than those proposed so far.

Authors:Jiyong Li; Xianfen Wang Abstract: Publication date: Available online 17 November 2016 Source:Applied Numerical Mathematics Author(s): Jiyong Li, Xianfen Wang In this paper, multi-step Runge–Kutta–Nyström methods for the numerical integration of special second-order initial value problems are proposed and studied. These methods include classical Runge–Kutta–Nyström methods as special cases. General order conditions are derived by using the theory of B-series based on the set of special Nyström-trees, and two explicit methods with order five and six, respectively, are constructed. Numerical results show that our new methods are more efficient in comparison with classical Runge–Kutta–Nyström methods and other well-known high quality methods proposed in the scientific literature.

Authors:Sarah W. Gaaf; Valeria Simoncini Abstract: Publication date: Available online 5 November 2016 Source:Applied Numerical Mathematics Author(s): Sarah W. Gaaf, Valeria Simoncini Given a large square matrix A and a sufficiently regular function f so that f ( A ) is well defined, we are interested in the approximation of the leading singular values and corresponding left and right singular vectors of f ( A ) , and in particular in the approximation of ‖ f ( A ) ‖ , where ‖ ⋅ ‖ is the matrix norm induced by the Euclidean vector norm. Since neither f ( A ) nor f ( A ) v can be computed exactly, we introduce a new inexact Golub–Kahan–Lanczos bidiagonalization procedure, where the inexactness is related to the inaccuracy of the operations f ( A ) v , f ( A ) ⁎ v . Particular outer and inner stopping criteria are devised so as to cope with the lack of a true residual. Numerical experiments with the new algorithm on typical application problems are reported.

Authors:Veselina K. Kyncheva; Viktor V. Yotov; Stoil I. Ivanov Abstract: Publication date: Available online 29 October 2016 Source:Applied Numerical Mathematics Author(s): Veselina K. Kyncheva, Viktor V. Yotov, Stoil I. Ivanov In this paper, we provide a local convergence analysis of Newton, Halley and Chebyshev iterative methods considered as methods for simultaneous determination of all multiple zeros of a polynomial f over an arbitrary normed field K . Convergence theorems with a priori and a posteriori error estimates for each of the proposed methods are established. The obtained results for Newton and Chebyshev methods are new even in the case of simple zeros. Three numerical examples are given to compare the convergence properties of the considered methods and to confirm the theoretical results.

Authors:Laura Gori; Francesca Pitolli Abstract: Publication date: Available online 11 October 2016 Source:Applied Numerical Mathematics Author(s): Laura Gori, Francesca Pitolli We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M = 4 and M = 5 are also given.

Authors:Zhongqiang Zhang; Heping Ma Abstract: Publication date: Available online 8 October 2016 Source:Applied Numerical Mathematics Author(s): Zhongqiang Zhang, Heping Ma We introduce a class of explicit balanced schemes for stochastic differential equations with coefficients of superlinearly growth satisfying a global monotone condition. The first scheme is a balanced Euler scheme and is of order half in the mean-square sense whereas it is of order one under additive noise. The second scheme is a balanced Milstein scheme, which is of order one in the mean-square sense. Some numerical results are presented.

Authors:Jingyang Guo; Jae-Hun Jung Abstract: Publication date: Available online 11 October 2016 Source:Applied Numerical Mathematics Author(s): Jingyang Guo, Jae-Hun Jung Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods are efficient high order numerical methods for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations. The original ENO and WENO methods are based on the polynomial interpolation and the overall convergence rate is uniquely determined by the total number of interpolation points involved for the approximation. In this paper, we propose non-polynomial ENO and WENO finite volume methods in order to enhance the local accuracy and convergence. The infinitely smooth radial basis functions (RBFs) are adopted as a non-polynomial interpolation basis. Particularly we use the multi-quadratic and Gaussian RBFs. The non-polynomial interpolation such as the RBF interpolation offers the flexibility to control the local error by optimizing the free parameter. Then we show that the non-polynomial interpolation can be represented as a perturbation of the polynomial interpolation. To guarantee the essentially non-oscillatory property, the monotone polynomial interpolation method is introduced as a switching method to the polynomial reconstruction adaptively near the non-smooth area. The numerical results show that the developed non-polynomial ENO and WENO methods with the monotone polynomial interpolation method enhance the local accuracy and give sharper solution profile.

Authors:G.Yu. Kulikov; M.V. Kulikova Abstract: Publication date: Available online 3 October 2016 Source:Applied Numerical Mathematics Author(s): G.Yu. Kulikov, M.V. Kulikova This paper further advances the idea of accurate Gaussian filtering towards efficient cubature Kalman filters for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the moment differential equations describing evolution of the predicted mean and covariance of the propagated Gaussian density in time are solved accurately, i.e. with negligible error. The latter allows the total error of the cubature Kalman filtering to be reduced significantly and results in a new accurate continuous-discrete cubature Kalman filtering method. At the same time, we revise the earlier developed version of the accurate continuous-discrete extended Kalman filter by amending the involved iteration and relaxing the utilized global error control mechanism. In addition, we build a mixed-type method, which unifies the best features of the accurate continuous-discrete extended and cubature Kalman filters. More precisely, the time updates are done in this state estimator as those in the first filter whereas the measurement updates are conducted with use of the third-degree spherical-radial cubature rule applied for approximating the arisen Gaussian-weighted integrals. All these are examined in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, and compared to the state-of-the-art cubature Kalman filters.

Authors:Fidalgo Abstract: Publication date: January 2017 Source:Applied Numerical Mathematics, Volume 111 Author(s): U. Fidalgo We use a connection between interpolatory quadrature formulas and Fourier series to find a wide class of convergent schemes of interpolatory quadrature rules. In the process we use techniques coming from Riemann–Hilbert problems for varying measures and convex analysis.

Authors:Alemdar Hasanov; Balgaisha Mukanova Abstract: Publication date: Available online 13 September 2016 Source:Applied Numerical Mathematics Author(s): Alemdar Hasanov, Balgaisha Mukanova Inverse problem of identifying the unknown spacewise dependent source F ( x ) in 1D wave equation u t t = c 2 u x x + F ( x ) G ( t ) + h ( x , t ) , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , T ) , from the Neumann-type measured output g ( t ) : = u x ( 0 , t ) is investigated. Most studies have attempted to reconstruct an unknown spacewise dependent source F ( x ) from the final observation u T ( x ) : = u ( x , T ) . Since a boundary measured data is most feasible from an engineering viewpoint, the identification problem has wide applications, in particular, in electrical networks governed by harmonically varying source for the linear wave equation u t t − u x x = F ( x ) c o s ( ω t ) , where ω > 0 is the frequency and F ( x ) is an unknown source term. In this paper Fourier Collocation Algorithm for reconstructing the spacewise dependent source F ( x ) is developed. This algorithm is based on Fourier expansion of the direct problem solution applied to the minimization problem for Tikhonov functional, by taking then a partial N-sum of the Fourier expansion. Tikhonov regularization is then applied to the obtained discrete ill-posed problem. To obtain high quality reconstruction in large values of the noise level, a numerical filtering algorithm is used for smoothing the noisy data. As an application, we demonstrate the ability of the algorithm on benchmark problems, in particular, on source identification problem in electrical networks governed by mono-frequency source. Numerical results show that the proposed algorithm allows to reconstruct the spacewise dependent source F ( x ) with enough high accuracy, in the presence of high noise levels.

Authors:Martin Bourne; Joab Winkler Abstract: Publication date: Available online 6 September 2016 Source:Applied Numerical Mathematics Author(s): Martin Bourne, Joab R. Winkler, Su Yi This paper considers the computation of the degree t of an approximate greatest common divisor d ( y ) of two Bernstein polynomials f ( y ) and g ( y ) , which are of degrees m and n respectively. The value of t is computed from the QR decomposition of the Sylvester resultant matrix S ( f , g ) and its subresultant matrices S k ( f , g ) , k = 2 , … , min ( m , n ) , where S 1 ( f , g ) = S ( f , g ) . It is shown that the computation of t is significantly more complicated than its equivalent for two power basis polynomials because (a) S k ( f , g ) can be written in several forms that differ in the complexity of the computation of their entries, (b) different forms of S k ( f , g ) may yield different values of t, and (c) the binomial terms in the entries of S k ( f , g ) may cause the ratio of its entry of maximum magnitude to its entry of minimum magnitude to be large, which may lead to numerical problems. It is shown that the QR decomposition and singular value decomposition (SVD) of the Sylvester matrix and its subresultant matrices yield better results than the SVD of the Bézout matrix, and that f ( y ) and g ( y ) must be processed before computations are performed on these resultant and subresultant matrices in order to obtain good results.