Authors:Qingtang Jiang; Dale K. Pounds Pages: 1 - 18 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 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:Wenting Shao; Xionghua Wu; Cheng Wang Pages: 19 - 32 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Wenting Shao, Xionghua Wu, Cheng Wang It is known that spectral methods offer exponential convergence for infinitely smooth solutions. However, they are not applicable for problems presenting singularities or thin layers, especially true for the ones with the location of singularity unknown. An adaptive domain decomposition method (DDM) integrated with Chebyshev tau method based on the highest derivative (CTMHD) is introduced to solve singular perturbed boundary value problems (SPBVPs). The proposed adaptive algorithm uses the refinement indicators based on Chebyshev coefficients to determine which subintervals need to be refined. Numerical experiments have been conducted to demonstrate the superior performance of the method for SPBVPs with a number of singularities including boundary layers, interior layers and dense oscillations. A fourth order nonlinear SPBVP is also concerned. The numerical results illustrate the efficiency and applicability of our adaptive algorithm to capture the locations of singularities, and the higher accuracy in comparison with some existing numerical methods in the literature.

Authors:Yasmina Daikh; Driss Yakoubi Pages: 33 - 49 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Yasmina Daikh, Driss Yakoubi We consider a variational formulation of the three dimensional Navier–Stokes equations provided with mixed boundary conditions. We write this formulation with three independent unknowns: the vorticity, the velocity and the pressure. Next, we propose a discretization by spectral methods. A detailed numerical analysis leads to a priori error estimates for the three unknowns.

Authors:M. Ableidinger; E. Buckwar Pages: 50 - 63 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): M. Ableidinger, E. Buckwar In this article we construct weak Runge–Kutta Munthe-Kaas methods for a finite-dimensional version of the stochastic Landau–Lifshitz equation (LL-equation). We formulate a Lie group framework for the stochastic LL-equation and derive regularity conditions for the corresponding SDE system on the Lie algebra. Using this formulation we define weak Munthe-Kaas methods based on weak stochastic Runge–Kutta methods (SRK methods) and provide sufficient conditions such that the Munthe-Kaas methods inherit the convergence order of the underlying SRK method. The constructed methods are fully explicit and preserve the norm constraint of the LL-equation exactly. Numerical simulations are provided to illustrate the convergence order as well as the long time behaviour of the proposed methods.

Authors:I. Alonso-Mallo; B. Cano; N. Reguera Pages: 64 - 74 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): I. Alonso-Mallo, B. Cano, N. Reguera In this paper, a thorough analysis is given for the order which is observed when integrating evolutionary linear partial differential equations with Lawson methods. The analysis is performed under the general framework of C0-semigroups in Banach spaces and hence it can be applied to the numerical time integration of many initial boundary value problems which are described by linear partial differential equations. Conditions of regularity and annihilation at the boundary of these problems are then stated to justify the precise order which is observed, including fractional order of convergence.

Authors:Pengzhan Huang Pages: 75 - 86 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Pengzhan Huang An efficient two-level finite element algorithm for solving the natural convection equations is developed and studied in this paper. By solving one small nonlinear system on a coarse mesh H and two large linearized problems on a fine mesh h = O ( H 7 − ε 2 ) with different loads, we can obtain an approximation solution ( u h , p h , T h ) with the convergence rate of same order as the usual finite element solution, which involves one large nonlinear natural convection system on the same fine mesh h. Furthermore, compared with the results of Si's algorithm in 2011, the given algorithm costs less computed time to get almost the same precision.

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:Jehanzeb H. Chaudhry; J.B. Collins; John N. Shadid Pages: 36 - 49 Abstract: Publication date: July 2017 Source:Applied Numerical Mathematics, Volume 117 Author(s): Jehanzeb H. Chaudhry, J.B. Collins, John N. Shadid Implicit–Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity-of-interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. The use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error in a quantity-of-interest.

Authors:Wansheng Wang Pages: 50 - 68 Abstract: Publication date: July 2017 Source:Applied Numerical Mathematics, Volume 117 Author(s): Wansheng Wang Motivated by recent stability results on one-step methods, especially Runge–Kutta methods, for the generalized pantograph equation (GPE), in this paper we study the stability of one-leg multistep methods for these equations since the one-leg methods have less computational cost than Runge–Kutta methods. To do this, a new stability concept, G q ( q ¯ ) -stability defined for variable stepsizes one-leg methods with the stepsize ratio q which is an extension of G-stability defined for constant stepsizes one-leg methods, is introduced. The Lyapunov functional of linear system is obtained and numerically approximated. It is proved that a G q ( q ¯ ) -stable fully-geometric mesh one-leg method can preserve the decay property of the Lyapunov functional for any q ∈ [ 1 , q ¯ ] . The asymptotic contractivity, a new stability concept at vanishing initial interval, is introduced for investigating the effect of the initial interval approximation on the stability of numerical solutions. This property and the bounded stability of G q ( q ¯ ) -stable one-leg methods for linear and nonlinear problems are analyzed. A numerical example which further illustrates our theoretical results is provided.

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:G.V. Kozyrakis; A.I. Delis; N.A. Kampanis Pages: 275 - 298 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): G.V. Kozyrakis, A.I. Delis, N.A. Kampanis Modern CFD applications require the treatment of general complex domains to accurately model the emerging flow patterns. In the present work, a new low order finite difference scheme is employed and tested for the numerical solution of the incompressible Navier–Stokes equations in a complex domain described in curvilinear coordinates. A staggered grid discretization is used on both the physical and computational domains. A subgrid based computation of the Jacobian and the metric coefficients of the transformation is used. The incompressibility condition, properly transformed in curvilinear coordinates, is enforced by an iterative procedure employing either a modified local pressure correction technique or the globally defined numerical solution of a general elliptic BVP. Results obtained by the proposed overall solution technique, exhibit very good agreement with other experimental and numerical calculations for a variety of domains and grid configurations. The overall numerical solver effectively treats the general complex domains.

Authors:Carlos J.S. Alves; Nuno F.M. Martins; Svilen S. Valtchev Pages: 299 - 313 Abstract: Publication date: May 2017 Source:Applied Numerical Mathematics, Volume 115 Author(s): Carlos J.S. Alves, Nuno F.M. Martins, Svilen S. Valtchev Two meshfree methods are developed for the numerical solution of the non-homogeneous Cauchy–Navier equations of elastodynamics in an isotropic material. The two approaches differ upon the choice of the basis functions used for the approximation of the unknown wave amplitude. In the first case, the solution is approximated in terms of a linear combination of fundamental solutions of the Navier differential operator with different source points and test frequencies. In the second method the solution is approximated by superposition of acoustic waves, i.e. fundamental solutions of the Helmholtz operator, with different source points and test frequencies. The applicability of the two methods is justified in terms of density results and a convergence result is proven. The accuracy of the methods is illustrated through 2D numerical examples. Applications to interior elastic wave scattering problems are also presented.

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.

Authors:Saifon Chaturantabut Abstract: Publication date: Available online 7 March 2017 Source:Applied Numerical Mathematics Author(s): Saifon Chaturantabut This work presents a model reduction framework using a temporal localized basis approach to efficiently reduce the simulation time for nonlinear dynamical systems with rapid changes over a short time period, and proposes a corresponding a priori error bound. This framework employs the proper orthogonal decomposition (POD) to construct localized basis sets from different temporal subdomains, which can be used in the Galerkin projection to accurately capture the important local dynamics of the system. The discrete empirical interpolation method (DEIM) with the corresponding temporal localized basis sets is then applied to efficiently compute the projected nonlinear terms. A heuristic procedure for subdividing snapshots over the temporal domain is proposed. This procedure first partitions the set of snapshots where there are possible significant changes in system dynamics, and then uses the notion of distance between subspaces to later remove unnecessary partitioning. An a priori error bound is derived to confirm the convergence of this framework and to explain how the propagated errors from the localized reduced systems affect the overall accuracy. Numerical experiments demonstrate the accuracy improvement of the temporal localized framework through a parametrized nonlinear miscible flow simulation. The results show the applicability of the proposed approach to various parameter values that are not necessary used for generating the POD and DEIM localized basis sets.

Authors:Chokri Chniti Abstract: Publication date: Available online 7 March 2017 Source:Applied Numerical Mathematics Author(s): Chokri Chniti The aim of this paper is to derive an appropriate second order transmission boundary conditions near the corner used in domain decomposition methods to study the reaction-diffusion problems (“ − ∇ . ( ν ( x ) ∇ . ) + η ( x ) . ”) with strong heterogeneity in the coefficients in a singular non-convex domain with Neumann and Dirichlet boundary condition. These transmission condition will be tested and compared numerically with other approaches.

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: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.