Authors:Kazuho Ito Pages: 1 - 20 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Kazuho Ito An energy conserving spectral scheme is presented for approximating the smooth solution of the dynamic elastica with free ends. The spatial discretization of the elastica is done on the basis of Galerkin spectral methods with a Legendre grid. It is established that the scheme has the unique solution and enjoys a spectral accuracy with respect to the size of the spatial grid. Moreover, some results of a numerical simulation are given to verify that the implemented scheme preserves the discrete energy.

Authors:Mariantonia Cotronei; Nada Sissouno Pages: 21 - 34 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Mariantonia Cotronei, Nada Sissouno The aim of the paper is to present Hermite-type multiwavelets, i.e. wavelets acting on vector data representing function values and consecutive derivatives, which satisfy the vanishing moment property with respect to elements in the space spanned by exponentials and polynomials. Such functions satisfy a two-scale relation which is level-dependent as well as the corresponding multiresolution analysis. An important feature of the associated filters is the possibility of factorizing their symbols in terms of the so-called cancellation operator. This is shown, in particular, in the situation where Hermite multiwavelets are obtained by completing interpolatory level-dependent Hermite subdivision operators, reproducing polynomial and exponential data, to biorthogonal systems. A few constructions of families of multiwavelet filters of this kind are proposed.

Authors:Peyman Hessari; Byeong-Chun Shin Pages: 35 - 52 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Peyman Hessari, Byeong-Chun Shin The subject of this paper is to investigate the first order system least squares Legendre and Chebyshev pseudo-spectral methods for coupled Stokes–Darcy equations. By introducing strain tensor as a new variable, Stokes–Darcy equations recast into a system of first order differential equations. The least squares functional is defined by summing up the weighted L 2 -norm of residuals of the first order system for coupled Stokes–Darcy equations. To treat Beavers–Joseph–Saffman interface conditions, the weighted L 2 -norm of these conditions are also added to the least squares functional. Continuous and discrete homogeneous functionals are shown to be equivalent to the combination of weighted H ( div ) and H 1 -norm for Stokes–Darcy equations. The spectral convergence for the Legendre and Chebyshev methods are derived. To demonstrate this analysis, numerical experiments are also presented.

Authors:Yuan-Ming Wang Pages: 53 - 67 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Yuan-Ming Wang This paper is concerned with a compact finite difference method with non-isotropic mesh sizes for a two-dimensional fourth-order nonlinear elliptic boundary value problem. By the discrete energy analysis, the optimal error estimates in the discrete L 2 , H 1 and L ∞ norms are obtained without any constraint on the mesh sizes. The error estimates show that the compact finite difference method converges with the convergence rate of fourth-order. Based on a high-order approximation of the solution, a Richardson extrapolation algorithm is developed to make the final computed solution sixth-order accurate. Numerical results demonstrate the high-order accuracy of the compact finite difference method and its extrapolation algorithm in the discrete L 2 , H 1 and L ∞ norms.

Authors:Jialin Hong; Lihai Ji; Linghua Kong; Tingchun Wang Pages: 68 - 81 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Jialin Hong, Lihai Ji, Linghua Kong, Tingchun Wang It has been pointed out in literature that the symplectic scheme of a nonlinear Hamiltonian system can not preserve the total energy in the discrete sense Ge and Marsden (1988) [10]. Moreover, due to the difficulty in obtaining a priori estimate of the numerical solution, it is very hard to establish the optimal error bound of the symplectic scheme without any restrictions on the grid ratios. In this paper, we develop and analyze a compact scheme for solving nonlinear Schrödinger equation. We introduce a cut-off technique for proving optimal L ∞ error estimate for the compact scheme. We show that the convergence of the compact scheme is of second order in time and of fourth order in space. Meanwhile, we define a new type of energy functional by using a recursion relationship, and then prove that the compact scheme is mass and energy-conserved, symplectic-conserved, unconditionally stable and can be computed efficiently. Numerical experiments confirm well the theoretical analysis results.

Authors:Xin-He Miao; Jian-Tao Yang; B. Saheya; Jein-Shan Chen Pages: 82 - 96 Abstract: Publication date: October 2017 Source:Applied Numerical Mathematics, Volume 120 Author(s): Xin-He Miao, Jian-Tao Yang, B. Saheya, Jein-Shan Chen In this paper, we consider the smoothing Newton method for solving a type of absolute value equations associated with second order cone (SOCAVE for short), which is a generalization of the standard absolute value equation frequently discussed in the literature during the past decade. Based on a class of smoothing functions, we reformulate the SOCAVE as a family of parameterized smooth equations, and propose the smoothing Newton algorithm to solve the problem iteratively. Moreover, the algorithm is proved to be locally quadratically convergent under suitable conditions. Preliminary numerical results demonstrate that the algorithm is effective. In addition, two kinds of numerical comparisons are presented which provides numerical evidence about why the smoothing Newton method is employed and also suggests a suitable smoothing function for future numerical implementations. Finally, we point out that although the main idea for proving the convergence is similar to the one used in the literature, the analysis is indeed more subtle and involves more techniques due to the feature of second-order cone.

Authors:Haibiao Zheng; Jiaping Yu; Li Shan Pages: 1 - 17 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Haibiao Zheng, Jiaping Yu, Li Shan The unconditional convergence of finite element method for two-dimensional time-dependent viscoelastic flow with an Oldroyd B constitutive equation is given in this paper, while all previous works require certain time-step restrictions. The approximation is stabilized by using the Discontinuous Galerkin (DG) approximation for the constitutive equation. The analysis bases on a splitting of the error into two parts: the error from the time discretization of the PDEs and the error from the finite element approximation of corresponding iterated time-discrete PDEs. The approach used in this paper can be applied to more general couple nonlinear parabolic and hyperbolic systems.

Authors:Wei Jiang; Na Liu Pages: 18 - 32 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Wei Jiang, Na Liu In this article, we proposed a new numerical method to obtain the approximation solution for the time variable fractional order mobile–immobile advection–dispersion model based on reproducing kernel theory and collocation method. The equation is obtained from the standard advection–dispersion equation (ADE) by adding the Coimbra's variable fractional derivative in time of order γ ( x , t ) ∈ [ 0 , 1 ] . In order to solve this kind of equation, we discuss and derive the ε-approximate solution in the form of series with easily computable terms in the bivariate spline space. At the same time, the stability and convergence of the approximation are investigated. Finally, numerical examples are provided to show the accuracy and effectiveness.

Authors:M. Król; M.V. Kutniv; O.I. Pazdriy Pages: 33 - 50 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): M. Król, M.V. Kutniv, O.I. Pazdriy The three-point difference schemes of high order accuracy for the numerical solving boundary value problems on a semi-infinite interval for systems of second order nonlinear ordinary differential equations with a not self-conjugate operator are constructed and justified. We proved the existence and uniqueness of solutions of the three-point difference schemes and obtained the estimate of their accuracy. The results of numerical experiments which confirm the theoretical results are given.

Authors:Mehdi Dehghan; Mostafa Abbaszadeh Pages: 51 - 66 Abstract: Publication date: September 2017 Source:Applied Numerical Mathematics, Volume 119 Author(s): Mehdi Dehghan, Mostafa Abbaszadeh In the current manuscript, we consider a fractional partial integro-differential equation that is called fractional evolution equation. The fractional evolution equation is based on the Riemann–Liouville fractional integral. The presented numerical algorithm is based on the following procedures: at first a difference scheme has been used to discrete the temporal direction and secondly the spectral element method is applied to discrete the spatial direction and finally these procedures are combined to obtain a full-discrete scheme. For the constructed numerical technique, we prove the unconditional stability and also obtain an error bound. We use the energy method to analysis the full-discrete scheme. We employ some test problems to show the high accuracy of the proposed technique. Also, we compare the obtained numerical results using the present method with the existing methods in the literature.

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:S. Magura; S. Petropavlovsky; S. Tsynkov; E. Turkel Pages: 87 - 116 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): S. Magura, S. Petropavlovsky, S. Tsynkov, E. Turkel Standard numerical methods often fail to solve the Helmholtz equation accurately near reentrant corners, since the solution may become singular. The singularity has an inhomogeneous contribution from the boundary data near the corner and a homogeneous contribution that is determined by boundary conditions far from the corner. We present a regularization algorithm that uses a combination of analytical and numerical tools to distinguish between these two contributions and ultimately subtract the singularity. We then employ the method of difference potentials to numerically solve the regularized problem with high-order accuracy over a domain with a curvilinear boundary. Our numerical experiments show that the regularization successfully restores the design rate of convergence.

Authors:Meng Li; Chengming Huang; Nan Wang Pages: 131 - 149 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Meng Li, Chengming Huang, Nan Wang In this paper, we are concerned with the numerical solution of the nonlinear fractional Ginzburg–Landau equation. Galerkin finite element method is used for the spatial discretization, and an implicit midpoint difference method is employed for the temporal discretization. The boundedness, existence and uniqueness of the numerical solution, and the unconditional error estimates in the L 2 -norm are investigated in details. To numerically solve the nonlinear system, linearized iterative algorithms are also considered. Finally, some numerical examples are presented to illustrate the effectiveness of the algorithm.

Authors:David A. Brown; David W. Zingg Pages: 150 - 181 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): David A. Brown, David W. Zingg Homotopy continuation, in combination with a quasi-Newton method, can be an efficient and robust technique for solving large sparse systems of nonlinear equations. The homotopy itself is pivotal in determining the efficiency and robustness of the continuation algorithm. As the homotopy is defined implicitly by a nonlinear system of equations to which the analytical solution is by assumption unknown, many properties of the homotopy can only be studied using numerical methods. The properties of a given homotopy which have the greatest impact on the corresponding continuation algorithm are traceability and linear solver performance. Metrics are presented for the analysis and characterization of these properties. Several homotopies are presented and studied using these metrics in the context of a parallel implicit three-dimensional Newton–Krylov–Schur flow solver for computational fluid dynamics. Several geometries, grids, and flow types are investigated in the study. Additional studies include the impact of grid refinement and the application of a coordinate transformation to the homotopy as measured through the traceability and linear solver performance metrics.

Authors:Hugh A. Carson; David L. Darmofal; Marshall C. Galbraith; Steven R. Allmaras Pages: 182 - 202 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Hugh A. Carson, David L. Darmofal, Marshall C. Galbraith, Steven R. Allmaras In this paper, we develop a priori estimates for the convergence of outputs, output error estimates, and localizations of output error estimates for Galerkin finite element methods. Output error estimates for order p finite element solutions are constructed using the Dual-Weighted Residual (DWR) method with a higher-order p ′ > p dual solution. Specifically, we analyze these DWR estimates for Continuous Galerkin (CG), Discontinuous Galerkin (DG), and Hybridized DG (HDG) methods applied to the Poisson problem. For all discretizations, as h → 0 , we prove that the output and output error estimate converge at order 2p and 2 p ′ (assuming sufficient smoothness), while localizations of the output and output error estimate converge at 2 p + d and p + p ′ + d . For DG, the results use a new post processing for the error associated with the lifting operator. For HDG, these rates improve an additional order when the stabilization is based upon an O ( 1 ) length scale.

Authors:M. Saedshoar Heris; M. Javidi Pages: 203 - 220 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): M. Saedshoar Heris, M. Javidi In this paper, fractional backward differential formulas (FBDF) are presented for the numerical solution of fractional delay differential equations (FDDEs) of the form λ n 0 C D t α n y ( t ) + λ n − 1 0 C D t α n − 1 y ( t ) + ⋯ + λ 1 0 C D t α 1 y ( t ) + λ n + 1 y ( t − τ ) = f ( t ) , t ∈ [ 0 , T ] , where λ i ∈ R ( i = 1 , ⋯ , n + 1 ) , λ n + 1 ≠ 0 , 0 ⩽ α 1 < α 2 < ⋯ < α n < 1 , T > 0 , in Caputo sense. Our investigation is focused on stability properties of the numerical methods and we determine stability regions for the FDDEs. Also we find the Green's functions for this equation corresponding to periodic/anti-periodic conditions in terms of the functions of Mittag Leffler type. Numerical tests are presented to confirm the strength of the approach under investigation.

Authors:Behnam Soleimani; Oswald Knoth; Rüdiger Weiner Pages: 221 - 237 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Behnam Soleimani, Oswald Knoth, Rüdiger Weiner Differential equations with both stiff and nonstiff parts can be solved efficiently by implicit–explicit (IMEX) methods. There have been considered various approaches in the literature. In this paper we introduce IMEX peer methods. We show that the combination of s-stage explicit and implicit peer methods, both of order p, gives an IMEX peer method of the same order. We construct methods of order p = s for s = 3 , 4 , where we compute the free parameters numerically to give good stability with respect to fast-wave–slow-wave problems from weather prediction. We implement these methods with and without step size control. Tests and comparisons with other methods for problems mostly from weather prediction show the high potential of IMEX peer methods.

Authors:Gabriel R. Barrenechea; Petr Knobloch Pages: 238 - 248 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Gabriel R. Barrenechea, Petr Knobloch The group finite element formulation is a strategy aimed at speeding the assembly of finite element matrices for time-dependent problems. This process modifies the Galerkin matrix of the problem in a non-consistent way. This may cause a deterioration of both the stability and convergence of the method. In this paper we prove results for a group finite element formulation of a convection–diffusion–reaction equation showing that the stability of the original discrete problem remains unchanged under appropriate conditions on the data of the problem and on the discretization parameters. A violation of these conditions may lead to non-existence of solutions, as one of our main results shows. An analysis of the consistency error introduced by the group finite element formulation and its skew-symmetric variant is given.

Authors:A.S. Fatemion Aghda; Seyed Mohammad Hosseini; Mahdieh Tahmasebi Pages: 249 - 265 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): A.S. Fatemion Aghda, Seyed Mohammad Hosseini, Mahdieh Tahmasebi The delay Cox–Ingersoll–Ross (CIR) model is an important model in the financial markets. It has been proved that the solution of this model is non-negative and its pth moments are bounded. However, there is no explicit solution for this model. So, proposing appropriate numerical method for solving this model which preserves non-negativity and boundedness of the model's solution is very important. In this paper, we concentrate on the balanced implicit method (BIM) for this model and show that with choosing suitable control functions the BIM provides numerical solution that preserves non-negativity of solution of the model. Moreover, we show the pth moment boundedness of the numerical solution of the method and prove the convergence of the proposed numerical method. Finally, we present some numerical examples to confirm the theoretical results, and also application of BIM to compute some financial quantities.

Authors:V.G. Pimenov; A.S. Hendy Pages: 266 - 276 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): V.G. Pimenov, A.S. Hendy A numerical method for fractional order differential equations (FDEs) and constant or time-varying delayed fractional differential equations (FDDEs) is constructed. This method is of BDF-type which is based on the interval approximation of the true solution by truncated shifted Chebyshev series. This approach can be reformulated in an equivalent way as a Runge–Kutta method and its Butcher tableau is given. A detailed local and global truncating errors analysis is deduced for the numerical solutions of FDEs and FDDEs. Illustrative examples are included to demonstrate the validity and applicability of the proposed approach.

Authors:Hongchao Kang; Junjie Ma Pages: 277 - 291 Abstract: Publication date: August 2017 Source:Applied Numerical Mathematics, Volume 118 Author(s): Hongchao Kang, Junjie Ma In this paper we mainly focus on the quadrature rules and asymptotic expansions for two classes of highly oscillatory Bessel integrals with algebraic or logarithmic singularities. Firstly, by two transformations, we transfer them into the standard types on [ − 1 , 1 ] , and derive two useful asymptotic expansions in inverse powers of the frequency ω. Then, based on the two asymptotic expansions, two methods are presented, respectively. One is the so-called Filon-type method. The other is the more efficient Clenshaw-Curtis–Filon-type method, which can be implemented in O ( N log N ) operations, based on Fast Fourier Transform (FFT) and fast computation of the modified moments. Here, through large amount of calculation and analysis, we can construct two important recurrence relations for computing the modified moments accurately, based on the Bessel's equation and some properties of the Chebyshev polynomials. In particular, we also provide error analysis for these methods in inverse powers of the frequency ω. Furthermore, we prove directly from the presented error bounds that these methods share the advantageous property, that the larger the values of the frequency ω, the higher the accuracy. The efficiency and accuracy of the proposed methods are illustrated by numerical examples.

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:John C. Butcher; Raffaele D'Ambrosio Pages: 69 - 86 Abstract: Publication date: July 2017 Source:Applied Numerical Mathematics, Volume 117 Author(s): John C. Butcher, Raffaele D'Ambrosio Partitioned general linear methods possessing the G-symplecticity property are introduced. These are intended for the numerical solution of separable Hamiltonian problems and, as for multivalue methods in general, there is a potential for loss of accuracy because of parasitic solution growth. The solution of mechanical problems over extended time intervals often benefits from interchange symmetry as well as from symplectic behaviour. A special type of symmetry, known as interchange symmetry, is developed from a model Runge–Kutta case to a full multivalue case. Criteria are found for eliminating parasitic behaviour and order conditions are explored.

Authors:Hassan Majidian Pages: 87 - 102 Abstract: Publication date: July 2017 Source:Applied Numerical Mathematics, Volume 117 Author(s): Hassan Majidian Numerical approximation of a general class of one-dimensional highly oscillatory integrals over bounded intervals with exponential oscillators is considered. A Filon-type method based on modified Clenshaw–Curtis quadrature rules is developed and its stability is established when the stationary points of the oscillator function are all of order two. Also, an error estimate for the method is provided, which shows that the method is convergent as the number of Clenshaw–Curtis points increases, and the rate of convergence depends only on the Sobolev regularity of the integrand. Using some numerical experiments, the theoretical results are illustrated.

Authors:Jingjun Zhao; Yan Fan; Yang Xu Pages: 103 - 114 Abstract: Publication date: July 2017 Source:Applied Numerical Mathematics, Volume 117 Author(s): Jingjun Zhao, Yan Fan, Yang Xu The paper is concerned with the delay-dependent stability analysis of symmetric Runge–Kutta methods, which include the Gauss methods and the Lobatto IIIA, IIIB and IIIS methods, for the second order delay differential equations with three parameters. By using the root locus technique, the root locus curve is given and the numerical stability region of symmetric Runge–Kutta methods is obtained. It is proved that, under some conditions, the analytical stability region is contained in the numerical stability region. Numerical examples confirming the theoretical results are presented.

Authors:G. Colldeforns-Papiol; L. Ortiz-Gracia; C.W. Oosterlee Pages: 115 - 138 Abstract: Publication date: July 2017 Source:Applied Numerical Mathematics, Volume 117 Author(s): G. Colldeforns-Papiol, L. Ortiz-Gracia, C.W. Oosterlee The SWIFT method for pricing European-style options on one underlying asset was recently published and presented as an accurate, robust and highly efficient technique. The purpose of this paper is to extend the method to higher dimensions by pricing exotic option contracts, called rainbow options, whose payoff depends on multiple assets. The multidimensional extension inherits the properties of the one-dimensional method, being the exponential convergence one of them. Thanks to the nature of local Shannon wavelets basis, we do not need to rely on a-priori truncation of the integration range, we have an error bound estimate and we use fast Fourier transform (FFT) algorithms to speed up computations. We test the method for similar examples with state-of-the-art methods found in the literature, and we compare our results with analytical expressions when available.

Authors:Francesco Dell'Accio; Maria Italia Gualtieri; Stefano Serra Capizzano; Gerhard Wanner First page: 1 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): Francesco Dell'Accio, Maria Italia Gualtieri, Stefano Serra Capizzano, Gerhard Wanner

Authors:Lidia Aceto; Helmut Robert Malonek; Graça Tomaz Pages: 2 - 9 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): Lidia Aceto, Helmut Robert Malonek, Graça Tomaz Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy–Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras.

Authors:A. Aimi; M. Diligenti; M.L. Sampoli; A. Sestini Pages: 10 - 23 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): A. Aimi, M. Diligenti, M.L. Sampoli, A. Sestini The application of the Isogeometric Analysis (IgA) paradigm to Symmetric Galerkin Boundary Element Method (SGBEM) is investigated. In order to obtain a very flexible approach, the study is here developed by using non-polynomial spline functions to represent both the domain boundary and the approximate solution. The numerical comparison between IGA-SGBEM and both curvilinear and standard SGBEM approaches shows the general capability of the presented method to produce accurate approximate solutions with less degrees of freedom.

Authors:D.A. Bini; G. Latouche; B. Meini Pages: 24 - 36 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): D.A. Bini, G. Latouche, B. Meini We revisit the shift technique applied to Quasi-Birth and Death (QBD) processes He et al. (2001) [13] in functional form by bringing the attention to the existence and properties of canonical factorizations. To this regard, we prove new results concerning the solutions of the quadratic matrix equations associated with the QBD. These results find applications to the solution of the Poisson equation for QBDs.

Authors:Dario A. Bini; Stefano Massei; Leonardo Robol Pages: 37 - 46 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): Dario A. Bini, Stefano Massei, Leonardo Robol We provide effective algorithms for solving block tridiagonal block Toeplitz systems with m × m quasiseparable blocks, as well as quadratic matrix equations with m × m quasiseparable coefficients, based on cyclic reduction and on the technology of rank-structured matrices. The algorithms rely on the exponential decay of the singular values of the off-diagonal submatrices generated by cyclic reduction. We provide a formal proof of this decay in the Markovian framework. The results of the numerical experiments that we report confirm a significant speed up over the general algorithms, already starting with the moderately small size m ≈ 10 2 .

Authors:L. Bos; S. De Marchi; M. Vianello Pages: 47 - 56 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): L. Bos, S. De Marchi, M. Vianello For a ∈ Z > 0 d we let ℓ a ( t ) : = ( cos ( a 1 t ) , cos ( a 2 t ) , ⋯ , cos ( a d t ) ) denote an associated Lissajous curve. We study such Lissajous curves which have the quadrature property for the cube [ − 1 , 1 ] d that ∫ [ − 1 , 1 ] d p ( x ) d μ d ( x ) = 1 π ∫ 0 π p ( ℓ a ( t ) ) d t for all polynomials p ( x ) ∈ V where V is either the space of d-variate polynomials of degree at most m or else the d-fold tensor product of univariate polynomials of degree at most m. Here d μ d is the product Chebyshev measure (also the pluripotential equilibrium measure for the cube). Among such Lissajous curves with this property we study the ones for which max p ∈ V deg ( p ( ℓ a ( t ) ) ) is as small as possible. In the tensor product case we show that this is uniquely minimized by g : = ( 1 , ( m + 1 ) , ( m + 1 ) 2 , ⋯ , ( m + 1 ) d − 1 ) . In the case of m = 2 n we construct discrete hyperinterpolation formulas which are easily evaluated with, for example, the Chebfun system ([6]).

Authors:Claude Brezinski; Michela Redivo-Zaglia Pages: 57 - 63 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): Claude Brezinski, Michela Redivo-Zaglia In this paper, we show how to construct various extensions of Shanks transformation for functions in a vector space. They are aimed at transforming a function tending slowly to its limit when the argument tends to infinity into another function with better convergence properties. Their expressions as ratio of determinants and recursive algorithms for their implementation are given. A simplified form of one of them is derived. It allows us to obtain a convergence result for an important class of functions. An application to integrable systems is discussed.

Authors:Alessandro Buccini Pages: 64 - 81 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): Alessandro Buccini The nonstationary preconditioned iteration proposed in a recent work by Donatelli and Hanke appeared on IP can be seen as an approximated iterated Tikhonov method. Starting from this observation we extend the previous iteration in two directions: the introduction of a regularization operator different from the identity (e.g., a differential operator) and the projection into a convex set (e.g., the nonnegative cone). Depending on the application both generalizations can lead to an improvement in the quality of the computed approximations. Convergence results and regularization properties of the proposed iterations are proved. Finally, the new methods are applied to image deblurring problems and compared with the iteration in the original work and other methods with similar properties recently proposed in the literature.

Authors:Kevin Burrage; Angelamaria Cardone; Raffaele D'Ambrosio; Beatrice Paternoster Pages: 82 - 94 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 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.

Authors:R. Cavoretto; S. De Marchi; A. De Rossi; E. Perracchione; G. Santin Pages: 95 - 107 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): R. Cavoretto, S. De Marchi, A. De Rossi, E. Perracchione, G. Santin In this paper we propose a new stable and accurate approximation technique which is extremely effective for interpolating large scattered data sets. The Partition of Unity (PU) method is performed considering Radial Basis Functions (RBFs) as local approximants and using locally supported weights. In particular, the approach consists in computing, for each PU subdomain, a stable basis. Such technique, taking advantage of the local scheme, leads to a significant benefit in terms of stability, especially for flat kernels. Furthermore, an optimized searching procedure is applied to build the local stable bases, thus rendering the method more efficient.

Authors:Emiliano Cirillo; Kai Hormann; Jean Sidon Pages: 108 - 118 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): Emiliano Cirillo, Kai Hormann, Jean Sidon Floater–Hormann interpolants constitute a family of barycentric rational interpolants which are based on blending local polynomial interpolants of degree d. Recent results suggest that the k-th derivatives of these interpolants converge at the rate of O ( h d + 1 − k ) for k ≤ d as the mesh size h converges to zero. So far, this convergence rate has been proven for k = 1 , 2 and for k ≥ 3 under the assumption of equidistant or quasi-equidistant interpolation nodes. In this paper we extend these results and prove that Floater–Hormann interpolants and their derivatives converge at the rate of O ( h j d + 1 − k ) , where h j is the local mesh size, for any k ≥ 0 and any set of well-spaced nodes.

Authors:Costanza Conti; Mariantonia Cotronei; Tomas Sauer Pages: 119 - 128 Abstract: Publication date: June 2017 Source:Applied Numerical Mathematics, Volume 116 Author(s): Costanza Conti, Mariantonia Cotronei, Tomas Sauer Subdivision schemes are known to be useful tools for approximation and interpolation of discrete data. In this paper, we study conditions for the convergence of level-dependent Hermite subdivision schemes, which act on vector valued data interpreting their components as function values and associated consecutive derivatives. In particular, we are interested in schemes preserving spaces of polynomials and exponentials. Such preservation property assures the existence of a cancellation operator in terms of which it is possible to obtain a factorization of the subdivision operators at each level. With the help of this factorization, we provide sufficient conditions for the convergence of the scheme based on some contractivity assumptions on the associated difference scheme.

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.