Authors:Yueqiang Shang; Jin Qin Pages: 1 - 21 Abstract: Publication date: July 2017 Source:Applied Numerical Mathematics, Volume 117 Author(s): Yueqiang Shang, Jin Qin Based on two-grid discretizations, some parallel finite element variational multiscale algorithms for the steady incompressible Navier–Stokes equations at high Reynolds numbers are presented and compared. In these algorithms, a stabilized Navier–Stokes system is first solved on a coarse grid, and then corrections are calculated independently on overlapped fine grid subdomains by solving a local stabilized linear problem. The stabilization terms for the coarse and fine grid problems are based on two local Gauss integrations. Error bounds for the approximate solution are estimated. Algorithmic parameter scalings are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, these algorithms can yield an optimal rate of convergence. Numerical results are given to verify the theoretical predictions and demonstrate the effectiveness of the proposed algorithms.

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:Yanmei Liu; Yubin Yan; Monzorul Khan Pages: 200 - 213 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): Yanmei Liu, Yubin Yan, Monzorul Khan In this paper, we consider the discontinuous Galerkin time stepping method for solving the linear space fractional partial differential equations. The space fractional derivatives are defined by using Riesz fractional derivative. The space variable is discretized by means of a Galerkin finite element method and the time variable is discretized by the discontinuous Galerkin method. The approximate solution will be sought as a piecewise polynomial function in t of degree at most q − 1 , q ≥ 1 , which is not necessarily continuous at the nodes of the defining partition. The error estimates in the fully discrete case are obtained and the numerical examples are given.

Authors:Cui Li; Chengjian Zhang Pages: 214 - 224 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): Cui Li, Chengjian Zhang This paper deals with a class of functional differential equations with piecewise continuous arguments. Block boundary value methods (BBVMs) are extended to solve this class of equations. It is shown under the Lipschitz condition that the order of convergence of an extended block boundary value method coincides with its order of consistency. Moreover, we study the linear stability of the extended methods and give the corresponding asymptotical stability criterion. In the end, with several numerical examples, the theoretical results and the computational effectiveness of the methods are further illustrated.

Authors:Ioannis K. Argyros; Ramandeep Behl; S.S. Motsa Pages: 225 - 234 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): Ioannis K. Argyros, Ramandeep Behl, S.S. Motsa We present a semilocal and local convergence analysis of Newton's method on a Banach space with a convergence structure to locate zeros of operators. P. Meyer introduced the concept of a Banach space with a convergence structure. Using this setting, he presented a finer semilocal convergence analysis for Newton's method than in related studies using the real norm theory. In all these studies the operator involved as well as its Fréchet derivative is bounded above by the same bound-operator. In the present study, we introduce a second bound operator which is a special case of the bound-operator leading to tighter majorizing sequences for Newton's method. Using this more flexible combination of bound-operators, we improve the results in the earlier studies. In the semilocal case, we obtain under the same or weaker sufficient convergence conditions more precise error bounds on the distances involved and in the local case not considered in the earlier studies, we obtain a larger radius of convergence. This way we expand the applicability of Newton's method. Some numerical examples are also provided to show the superiority of the new results over the old results.

Authors:Qian Guo; Wei Liu; Xuerong Mao; Rongxian Yue Pages: 235 - 251 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): Qian Guo, Wei Liu, Xuerong Mao, Rongxian Yue The partially truncated Euler–Maruyama (EM) method is proposed in this paper for highly nonlinear stochastic differential equations (SDEs). We will not only establish the finite-time strong L r -convergence theory for the partially truncated EM method, but also demonstrate the real benefit of the method by showing that the method can preserve the asymptotic stability and boundedness of the underlying SDEs.

Authors:Jamal Amani Rad; Kourosh Parand Pages: 252 - 274 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): Jamal Amani Rad, Kourosh Parand The most recent update of financial option models is American options under stochastic volatility models with jumps in returns (SVJ) and stochastic volatility models with jumps in returns and volatility (SVCJ). To evaluate these options, mesh-based methods are applied in a number of papers but it is well-known that these methods depend strongly on the mesh properties which is the major disadvantage of them. Therefore, we propose the use of the meshless methods to solve the aforementioned options models, especially in this work we select and analyze one scheme of them, named local radial point interpolation (LRPI) based on Wendland's compactly supported radial basis functions (WCS-RBFs) with C 6 , C 4 and C 2 smoothness degrees. The LRPI method which is a special type of meshless local Petrov–Galerkin method (MLPG), offers several advantages over the mesh-based methods, nevertheless it has never been applied to option pricing, at least to the very best of our knowledge. These schemes are the truly meshless methods, because, a traditional non-overlapping continuous mesh is not required, neither for the construction of the shape functions, nor for the integration of the local sub-domains. In this work, the American option which is a free boundary problem, is reduced to a problem with fixed boundary using a Richardson extrapolation technique. Then the implicit–explicit (IMEX) time stepping scheme is employed for the time derivative. Numerical experiments are presented showing that the proposed approaches are extremely accurate and fast.

Authors:Sonia Seyed Allaei; Teresa Diogo; Magda Rebelo Pages: 2 - 17 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Sonia Seyed Allaei, Teresa Diogo, Magda Rebelo We consider a class of nonlinear singular Hammerstein Volterra integral equations. In general, these equations will have kernels containing both an end point and an Abel-type singularity, with exact solutions being typically nonsmooth. Under certain conditions, a uniformly convergent iterative solution is obtained on a small interval near the origin. In this work, two product integration methods are proposed and analyzed where the integral over a small initial interval is calculated analytically, allowing the optimal convergence rates to be achieved. This is illustrated by some numerical examples.

Authors:A. Cardone; R. D'Ambrosio; B. Paternoster Pages: 18 - 29 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): A. Cardone, R. D'Ambrosio, B. Paternoster The present paper illustrates the construction of direct quadrature methods of arbitrary order for Volterra integral equations with periodic solution. The coefficients of these methods depend on the parameters of the problem, following the exponential fitting theory. The convergence of these methods is analyzed, and some numerical experiments are illustrated to confirm theoretical expectations and for comparison with other existing methods.

Authors:Dajana Conte; Beatrice Paternoster Pages: 30 - 37 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Dajana Conte, Beatrice Paternoster The purpose of this paper is to employ graphics processing units for the numerical solution of large systems of weakly singular Volterra Integral Equations (VIEs), by means of Waveform Relaxation (WR) methods. A CUDA solver based on different kinds of WR iterations is developed. Numerical results on large systems of VIEs arising from the semi-discretization in space of fractional diffusion-wave equations are presented, showing the obtained speed-up.

Authors:L. Grammont; H. Kaboul; M. Ahues Pages: 38 - 46 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): L. Grammont, H. Kaboul, M. Ahues This paper deals with nonlinear Fredholm integral equations of the second kind. We study the case of a weakly singular kernel and we set the problem in the space L 1 ( [ a , b ] , C ) . As numerical method, we extend the product integration scheme from C 0 ( [ a , b ] , C ) to L 1 ( [ a , b ] , C ) .

Authors:Filomena D. d' Almeida; Rosário Fernandes Pages: 47 - 54 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Filomena D. d' Almeida, Rosário Fernandes For the solution of a weakly singular Fredholm integral equation of the 2nd kind defined on a Banach space, for instance L 1 ( [ a , b ] ) , the classical projection methods with the discretization of the approximating operator on a finite dimensional subspace usually use a basis of this subspace built with grids on [ a , b ] . This may require a large dimension of the subspace. One way to overcome this problem is to include more information in the approximating operator or to compose one classical method with one step of iterative refinement. This is the case of Kulkarni method or iterated Kantorovich method. Here we compare these methods in terms of accuracy and arithmetic workload. A theorem stating comparable error bounds for these methods, under very weak assumptions on the kernel, the solution and the space where the problem is set, is given.

Authors:Paulo B. Vasconcelos Pages: 55 - 62 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Paulo B. Vasconcelos Data-sparse representation techniques are emerging on computing approximate solutions for large scale problems involving matrices with low numerical rank. This representation provides both low memory requirements and cheap computational costs. In this work we consider the numerical solution of a large dimensional problem resulting from a finite rank discretization of an integral radiative transfer equation in stellar atmospheres. The integral operator, defined through the first exponential-integral function, is of convolution type and weakly singular. Hierarchically semiseparable representation of the matrix operator with low-rank blocks is built and data-sparse matrix computations can be performed with almost linear complexity. This representation of the original fully populated matrix is an algebraic multilevel structure built from a specific hierarchy of partitions of the matrix indices. Numerical tests illustrate the benefits of this matrix technique compared to standard storage schemes, dense and sparse, in terms of computational cost as well as memory requirements. This approach is particularly useful when a fine discretization of the integral equation is required and the resulting linear system of equations is of large dimension and numerically difficult to solve.

Authors:T. Diogo; P.M. Lima; A. Pedas; G. Vainikko Pages: 63 - 76 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): T. Diogo, P.M. Lima, A. Pedas, G. Vainikko This work is concerned with the construction and analysis of high order numerical methods for solving initial value problems for linear Volterra integro-differential equations with different types of singularities. Using an integral reformulation of the initial value problem, a smoothing transformation is applied so that the exact solution of the resulting equation does not contain any singularities in its derivatives up to a certain order. After that, the regularized equation is solved by a piecewise polynomial collocation method on a uniform or mildly graded grid. Finally, the obtained spline approximations can be used to define (typically non-polynomial) approximations for the initial value problem. The theoretical results are tested by some numerical examples.

Authors:Jana Burkotová; Irena Rachůnková; Ewa B. Weinmüller Pages: 77 - 96 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Jana Burkotová, Irena Rachůnková, Ewa B. Weinmüller In this paper, analytical properties of systems of singular linear ordinary differential equations with variable coefficient matrices and nonsmooth inhomogeneities are investigated. The aim is to precisely formulate conditions which are necessary and sufficient for the existence and uniqueness of solutions which are at least continuous on the closed interval including the singular point. Smoothness properties of such solutions are also discussed.

Authors:P.M. Lima; M.L. Morgado; M. Schöbinger; E.B. Weinmüller Pages: 97 - 107 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): P.M. Lima, M.L. Morgado, M. Schöbinger, E.B. Weinmüller We study the numerical solution of a singular free boundary problem for a second order nonlinear ordinary differential equation, where the differential operator is the degenerate m-Laplacian. A typical difficulty arising in free boundary problems is that the analytical solution may become non-smooth at one boundary or at both boundaries of the interval of integration. A numerical method proposed in [18] consists of two steps. First, a smoothing variable transformation is applied to the analytical problem in order to improve the smoothness of its solution. Then, the problem is discretized by means of a finite difference scheme. In the present paper, we consider an alternative numerical approach. We first transform the original problem into a special parameter dependent problem sometimes referred to as an ‘eigenvalue problem’. By applying a smoothing variable transformation to the resulting equation, we obtain a new problem whose solution is smoother, and so the open domain Matlab collocation code bvpsuite [17] can be successfully applied for its numerical approximation.

Authors:Maria Luísa Morgado; Magda Rebelo; Luis L. Ferrás; Neville J. Ford Pages: 108 - 123 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Maria Luísa Morgado, Magda Rebelo, Luis L. Ferrás, Neville J. Ford In this work we present a new numerical method for the solution of the distributed order time-fractional diffusion equation. The method is based on the approximation of the solution by a double Chebyshev truncated series, and the subsequent collocation of the resulting discretised system of equations at suitable collocation points. An error analysis is provided and a comparison with other methods used in the solution of this type of equation is also performed.

Authors:Hitoshi Mahara; Koshiro Mizobe; Katsuyuki Kida; Kazuaki Nakane Pages: 124 - 131 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Hitoshi Mahara, Koshiro Mizobe, Katsuyuki Kida, Kazuaki Nakane Boundary detection is a very important procedure in the image analysis of structures. Because vague boundaries are often traced by hand in a time-consuming process, an automated boundary detection system is needed. Here, we introduce a method to detect vague boundaries by solving a reaction–diffusion system. This method makes blurry boundaries clear, even if there are irregularities in image brightness. Combining our method with ordinary image analysis methods, we are able to derive useful information from images with vague boundaries. Because the algorithm of this method depends on mathematical theory, we can apply it to different kinds of images. To confirm the effectiveness of our method, two different kinds of images were tested, those of capillaries and prior austenite grain structures. Good numerical results were achieved for both types of images.

Authors:Maria Carmela De Bonis; Donatella Occorsio Pages: 132 - 153 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Maria Carmela De Bonis, Donatella Occorsio In this paper we propose a global method to approximate the derivatives of the weighted Hilbert transform of a given function f H p ( f w α , t ) = d p d t p ⨍ 0 + ∞ f ( x ) x − t w α ( x ) d x = p ! ⨎ 0 + ∞ f ( x ) ( x − t ) p + 1 w α ( x ) d x , where p ∈ { 1 , 2 , … } , t > 0 , and w α ( x ) = e − x x α is a Laguerre weight. The right-hand integral is defined as the finite part in the Hadamard sense. The proposed numerical approach is convenient when the approximation of the function H p ( f w α , t ) is required. Moreover, if there is the need, all the computations can be performed without differentiating the density function f. Numerical stability and convergence are proved in suitable weighted uniform spaces and numerical tests which confirm the theoretical estimates are presented.

Authors:J.A. Roberts; A. Al Themairi Pages: 154 - 164 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): J.A. Roberts, A. Al Themairi We introduce delay dynamics to an ordinary differential equation model of tumour growth based upon von Bertalanffy's growth model, a model which has received little attention in comparison to other models, such as Gompertz, Greenspan and logistic models. Using existing, previously published data sets we show that our delay model can perform better than delay models based on a Gompertz, Greenspan or logistic formulation. We look for replication of the oscillatory behaviour in the data, as well as a low error value (via a Least-Squares approach) when comparing. We provide the necessary analysis to show that a unique, continuous, solution exists for our model equation and consider the qualitative behaviour of a solution near a point of equilibrium.

Authors:Giuseppe Izzo; Zdzislaw Jackiewicz Pages: 165 - 178 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Giuseppe Izzo, Zdzislaw Jackiewicz In this paper we use the theoretical framework of General Linear Methods (GLMs) to analyze and generalize the class of Cash's Modified Extended Backward Differentiation Formulae (MEBDF). Keeping the structure of MEBDF and their computational cost we propose a more general class of methods that can be viewed as a composition of modified linear multistep methods. These new methods are characterized by smaller error constants and possibly larger angles of A ( α ) -stability. Numerical experiments which confirm the good performance of these methods on a set of stiff problems are also reported.

Authors:S. McKee; Jose A. Cuminato Pages: 179 - 187 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): S. McKee, Jose A. Cuminato A new integration technique, which is suitable for integrands with multiple weak singularities, is introduced. Local truncation errors are given. This scheme, when applied to the Beta function, is shown to emerge naturally from discrete fractional integration. To illustrate the effectiveness of the integration method a numerical example is provided, with somewhat unexpected convergence results.

Authors:Somayeh Gh. Bardeji; Isabel N. Figueiredo; Ercília Sousa Pages: 188 - 200 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Somayeh Gh. Bardeji, Isabel N. Figueiredo, Ercília Sousa An optical flow variational model is proposed for a sequence of images defined on a domain in R 2 . We introduce a regularization term given by the L 1 norm of a fractional differential operator. To solve the minimization problem we apply the split Bregman method. Extensive experimental results, with performance evaluation, are presented to demonstrate the effectiveness of the new model and method and to show that our algorithm performs favorably in comparison to another existing method. We also discuss the influence of the order α of the fractional operator in the estimation of the optical flow, for 0 ≤ α ≤ 2 . We observe that the values of α for which the method performs better depend on the geometry and texture complexity of the image. Some extensions of our algorithm are also discussed.

Authors:Zhiqiang Li; Yubin Yan; Neville J. Ford Pages: 201 - 220 Abstract: Publication date: April 2017 Source:Applied Numerical Mathematics, Volume 114 Author(s): Zhiqiang Li, Yubin Yan, Neville J. Ford In this paper, we first introduce an alternative proof of the error estimates of the numerical methods for solving linear fractional differential equations proposed in Diethelm [6] where a first-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and the convergence order of the proposed numerical method is O ( Δ t 2 − α ) , 0 < α < 1 , where α is the order of the fractional derivative and Δt is the step size. We then use a similar idea to prove the error estimates of the high order numerical method for solving linear fractional differential equations proposed in Yan et al. [37], where a second-degree compound quadrature formula was used to approximate the Hadamard finite-part integral and we show that the convergence order of the numerical method is O ( Δ t 3 − α ) , 0 < α < 1 . Numerical examples are given to show that the numerical results are consistent with the theoretical results.

Abstract: Publication date: Available online 16 February 2017 Source:Applied Numerical Mathematics Author(s): Kevin Burrage, Angelamaria Cardone, Raffaele D'Ambrosio, Beatrice Paternoster In this paper a general class of diffusion problem is considered, where the standard time derivative is replaced by a fractional one. For the numerical solution, a mixed method is proposed, which consists of a finite difference scheme through space and a spectral collocation method through time. The spectral method considerably reduces the computational cost with respect to step-by-step methods to discretize the fractional derivative. Some classes of spectral bases are considered, which exhibit different convergence rates and some numerical results based on time diffusion reaction diffusion equations are given.

Abstract: Publication date: Available online 16 February 2017 Source:Applied Numerical Mathematics Author(s): Qingtang Jiang, Dale K. Pounds Multiresolution techniques for (mesh-based) surface processing have been developed and successfully used in surface progressive transmission, compression and other applications. A triangular mesh allows 3 , dyadic and 7 refinements. The 3 -refinement is the most appealing one for multiresolution data processing since it has the slowest progression through scale and provides more resolution levels within a limited capacity. The 3 refinement has been used for surface subdivision and for discrete global grid systems Recently lifting scheme-based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing. If biorthogonal wavelets (with either dyadic or 3 refinement) have certain smoothness, they will have big supports. In other words, the corresponding multiscale algorithms have large templates; and this is undesirable for surface processing. On the other hand, frames provide a flexibility for the construction of system generators (called framelets) with high symmetry and smaller supports. In this paper we study highly symmetric 3 -refinement wavelet bi-frames for surface processing. We design the frame algorithms based on the vanishing moments and smoothness of the framelets. The frame algorithms obtained in this paper are given by templates so that one can easily implement them. We also present interpolatory 3 subdivision-based frame algorithms. In addition, we provide frame ternary multiresolution algorithms for boundary vertices on an open surface.

Authors:Darae Jeong; Junseok Kim Abstract: Publication date: Available online 3 February 2017 Source:Applied Numerical Mathematics Author(s): Darae Jeong, Junseok Kim We consider phase-field models and associated numerical methods for tissue growth. The model consists of the Cahn–Hilliard equation with a source term. In order to solve the equations accurately and efficiently, we propose a hybrid method based on an operator splitting method. First, we solve the contribution from the source term analytically and redistribute the increased mass around the tissue boundary position. Subsequently, we solve the Cahn–Hilliard equation using the nonlinearly gradient stable numerical scheme to make the interface transition profile smooth. We then perform various numerical experiments and find that there is a good agreement when these computational results are compared with analytic solutions.

Authors:Mahboub Baccouch Abstract: Publication date: Available online 18 January 2017 Source:Applied Numerical Mathematics Author(s): Mahboub Baccouch In this paper, we propose and analyze a superconvergent discontinuous Galerkin (DG) method for nonlinear second-order initial-value problems for ordinary differential equations. Optimal a priori error estimates for the solution and for the auxiliary variable that approximates the first-order derivative are derived in the L 2 -norm. The order of convergence is proved to be p + 1 , when piecewise polynomials of degree at most p are used. We further prove that the p-degree DG solutions are O ( h 2 p + 1 ) superconvergent at the downwind points. Finally, we prove that the DG solutions are superconvergent with order p + 2 to a particular projection of the exact solutions. The proofs are valid for arbitrary nonuniform regular meshes and for piecewise P p polynomials with arbitrary p ≥ 1 . Computational results indicate that the theoretical orders of convergence and superconvergence are optimal.

Authors:Huanhuan Yang; Alessandro Veneziani Abstract: Publication date: Available online 16 January 2017 Source:Applied Numerical Mathematics Author(s): Huanhuan Yang, Alessandro Veneziani Clinical oriented applications of computational electrocardiology require efficient and reliable identification of patient-specific parameters of mathematical models based on available measures. In particular, the estimation of cardiac conductivities in models of potential propagation is crucial, since they have major quantitative impact on the solution. Available estimates of cardiac conductivities are significantly diverse in the literature and the definition of experimental/mathematical estimation techniques is an open problem with important practical implications in clinics. We have recently proposed a methodology based on a variational procedure, where the reliability is confirmed by numerical experiments. In this paper we explore model-order-reduction techniques to fit the estimation procedure into timelines of clinical interest. Specifically we consider the Monodomain model and resort to Proper Orthogonal Decomposition (POD) techniques to take advantage of an off-line step when solving iteratively the electrocardiological forward model online. In addition, we perform the Discrete Empirical Interpolation Method (DEIM) to tackle the nonlinearity of the model. While standard POD techniques usually fail in this kind of problems, due to the wave-front propagation dynamics, an educated novel sampling of the parameter space based on the concept of Domain of Effectiveness introduced here dramatically reduces the computational cost of the inverse solver by at least 95%.

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: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: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: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: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: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: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: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: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.